Full text of "Human Knowledge - Its Scope And Limits"

See other formats

BERTRAND RUSSELL

HUMAN

KNOWLEDGE

Its Scope and Limits

LONDON

GEORGE ALLEN AND UNWIN LTD

Ruskin House, Museum Street

1923.

First Published
Reprinted .
Reprinted .
Reprinted .

Re printed .
Reprinted .
Reprinted .

Re printed ,
Reprinted .

1934

1938

1942

1944

1945

1946

1947

1948

PREFACE

T he following pages are addressed, not only or primarily
to professional philosophers, but to that much larger public
which is interested in philosophical questions without being
willing or able to devote more than a limited amount of time
to considering them. Descartes, Leibniz, Locke, Berkeley, and
Hume wrote for a public of this sort, and I think it is unfortunate
that during the last hundred and sixty years or so philosophy
has come to be regarded as almost as technical as mathematics.
Logic, it must be admitted, is technical in the same way as
mathematics is, but logic, I maintain, is not part of philosophy.
Philosophy proper deals with matters of interest to the general
educated public, and loses much of its value if only a few pro-
fessionals can understand what is said.

In this book I have sought to deal, as comprehensively as I
am able, with a very large question: how comes it that human
beings, whose contacts with the world are brief and personal
and limited, are nevertheless able to know as much as they do
know? Is the belief in our knowledge partly illusory? And, if
not, what must we know otherwise than through the senses?
Since I have dealt in earlier books with some parts of this problem,
I am compelled to repeat, in a larger context, discussions of
certain matters which I have considered elsewhere, but I have
reduced such repetition to the minimum compatible with my
purpose.

One of the difficulties of the subject with which I am concerned
is that we must employ words which are common in ordinary
speech, such as “belief”, “truth”, “knowledge”, and “percep-
tion”. Since these words, in their every-day uses, are vague and
unprecise, and since no precise words are ready to hand by
which to replace them, it is inevitable that everything said in
the earlier stages of our inquiry should be unsatisfactory from
the point of view that we hope to arrive at in the end. Our
increase of knowledge, assuming that we are successful, is like
that of a traveller approaching a mountain through a haze: at
first only certain large features are discernible, 'and even they
have indistinct boundaries, but gradually more detail becomes
visible and edges become sharper. So, in our discussions, it is

human knowledge: its scope and limits

impossible first to clear up one problem and then proceed to
another, for the intervening haze envelops all alike. At every
stage, though one part of our problem may be in the focus of
attention, all parts are more or less relevant. The different key
words that we must use are all interconnected, and so long as
some remain vague, others must, more or less, share this defect.
It follows that what is said at first is liable to require emendation
later VThe Prophet announced that if two texts of the Koran
appeared inconsistent, the later text was to be taken as authorita-
tive, and I should wish the reader to apply a similar principle in
'interpreting what is said in this book.

The book has been read in typescript by my friend and pupil
Mr. C. K. Hill, and I am indebted to him for many valuable
criticisms, suggestions, and emendations. Large parts of the
typescript have also been read by Mr. Hiram J. McLendon,
who has made a number of useful suggestions.

Part III, Chapter IV, on “Physics and Experience”, is a
reprint, with few alterations, of a little book with the above
title, published by the Cambridge University Press, to whom I
owe thanks for permission to reprint it.

CONTENTS

INTRODUCTION PAGE 9

PART I. THE WORLD OF SCIENCE

I Individual and Social Knowledge 17

II The Universe of Astronomy 23

III The World of Physics 29

IV Biological Evolution 43

V The Physiology of Sensation and Volition 51

VI The Science of Mind 57

PART II. LANGUAGE

I The Uses of Language 71

II Ostensive Definition 78

III Proper Names 87

IV Egocentric Particulars 100

V Suspended Reactions: Knowledge and Belief 109

VI Sentences 119

VII External Reference of Ideas and Beliefs 123

VIII Truth: Elementary Forms 127

IX Logical Words and Falsehood 136

X General Knowledge 146

XI Fact, Belief, Truth, and Knowledge 159

PART III. SCIENCE AND PERCEPTION
INTRODUCTION 177

I Knowledge of Facts and Knowledge of Laws 180

II Solipsism 191

III Probable Inference in Common-sense Practice 198

IV Physics and Experience 21 1

V Time in Experience / 226

VI Space in Psychology 233

VII Mind and Matter 240

HUMAN knowledge: its scope and limits

PART IV. SCIENTIFIC CONCEPTS

Interpretation

tPAGE 251

Minimum Vocabularies

259

III

Structure

267

Structure and Minimum Vocabularies

274

Time, Public and Private

284

Space in Classical Physics

295

VII

Space-Time

305

VIII

The Principle of Individuation

310

Causal Laws

326

Space-time and Causality

337

PART V. PROBABILITY

INTRODUCTION

353

Kinds of Probability

356

Mathematical Probability

362

III

The Finite-Frequency Theory

368

The Mises-Reichenbach Theory

380

Keynes’s Theory of Probability

390

Degrees of Credibility

398

VII

Probability and Induction

PART

VI. POSTULATES OF SCIENTIFIC INFERENCE

Kinds of Knowledge

439

The Role of Induction

45 i

III

The Postulate of Natural Kinds

456

Knowledge Transcending Experience

463

Causal Lines

47 i

Structure and Causal Laws

479

VII

Interaction

494

VIII

Analogy

501

Summary of Postulates

506

The Limits of Empiricism

516

Index 528

INTRODUCTION

T he central purpose of this book is to examine the relation
between individual experience and the general body of
scientific knowledge. It is taken for granted that scientific
knowledge, in its broad outlines, is to be accepted. Scepticism,
while logically impeccable, is psychologically impossible, and
there is an element of frivolous insincerity in any philosophy
which pretends to accept it. Moreover, if scepticism is to be
theoretically defensible it must reject all inferences from what
is experienced; a partial scepticism, such as the denial of physical
events experienced by no one, or a solipsism which allows events
in my future or in my unremembered past, has no logical
justification, since it must admit principles of inference which
lead to beliefs that it rejects.

Ever since Kant, or perhaps it would be more just to say
ever since Berkeley, there has been what I regard as a mistaken
tendency among philosophers to allow the description of the
world to be influenced unduly by considerations derived from
the nature of human knowledge. To scientific common sense
(which I accept) it is plain that only an infinitesimal part of the
universe is known, that there were countless ages during which
there was no knowledge, and that there probably will be countless
ages without knowledge in the future. Cosmically and causally,
knowledge is an unimportant feature of the universe; a science
which omitted to mention its occurrence might, from an im-
personal point of view, suffer only from a very trivial imperfection.
In describing the world, subjectivity is a vice. Kant spoke of
"himself as having effected a “Copernican revolution”, but he
would have been more accurate if he had spoken of a “Ptolemaic
counter-revolution”, since he put Man back at the centre from
which Copernicus had dethroned him.

But when we ask, not “what sort of world do we live in?”
but “how do we come by our knowledge about the world?”
subjectivity is in order. What each man knows is, in an important
sense, dependent upon his own individual experience: he knows
what he has seen and heard, what he has read and what he has
been told, and also what, from these data, he has been able to
infer. It is individual, not collective, experience that is here in

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

question, for an inference is required to pass from my data
to the acceptance of testimony. If I believe that there is such a
place as Semipalatinsk, I believe it because of things that have
happened to me\ and unless certain substantial principles of
inference are accepted, I shall have to admit that all these things
might have happened to me without there being any such place.

The desire to escape from subjectivity in the description of
the world (which I share) has led some modern philosophers
astray — at least so it seems to me — in relation to theory of
knowledge. Finding its problems distasteful, they have tried to
deny that these problems exists That da ta are private. And indi-
vidual is a thesis which has been familiar since the time of
Protagoras. This thesis has been denied because it has been
thought, as Protagoras thought, that, if admitted, it must lead
to the conclusion that all knowledge is private and individual.

( For my part, while I admit the thesis, I deny the conclusion;
how and why, the following pages are intended to show.

In virtue of certain events in my own life, I have a number
of beliefs about events that I do not experience — the thoughts
and feelings of other people, the physical objects that surround
me, the historical and geological past of the earth, and the remote
regions of the universe that are studied in astronomy. For my
part, I accept these beliefs as valid, apart from errors of detail.
By this acceptance I commit myself to the view that there are
valid processes of inference from events to other events — more
particularly, from events of which I am aware without inference
to events of which I have no such awareness. To discover what
these processes are is a matter of analysis of scientific and
common-sense procedure, in so far as such procedure is generally
accepted as scientifically valid.

Inference from a group of events to other events can only be
justified if the world has certain characteristics which are not
logically necessary. So far as deductive logic can show, any
collection of events might be the whole universe; if, then, I am
ever to be able to infer events, I must accept principles of inference
which lie outside deductive logic. All inference from events to
events demands some kind of interconnection between different
occurrences. Such interconnection is traditionally asserted in
the principle of causality or natural law. It is implied, as we
shall find, in whatever limited validity may be assigned to

INTRODUCTION

induction by simple enumeration. But the traditional ways of
formulating the kind of interconnection that must be postulated
are in many ways defective, some being too stringent and some
not sufficiently sdt To discover the minimum principles required toj
justify scientific inferences is one ofthe mam^purposes of this book j
' iris ’'^ifonimoilpace*' to say that the substantial inferences
of science, as opposed to those of logic and mathematics, are only
probable — that is to say, when the premisses are true and the
inference correct, the conclusion is only likely to be true. It is
therefore necessary to examine what is meant by “probability”.
It will be found that there are two different concepts that may
be meant. On the one hand, there is mathematical probability: if
a class has n members, and m of them have a certain characteristic,
the mathematical probability that an unspecified member of this
class will have the characteristic in question is m/n. On the other
hand, there is a wider and vaguer concept, which I call “degree of
credibility”, which is the amount of credence that it is rational to
assign to a more or less uncertain proposition. Both kinds of proba-
bility are involved in stating the principles of scientific inference.

The course of our inquiry, in broad outline, will be as follows.

Part I, on the world of science, describes some of the main
features of the universe which scientific investigation has made
probable. This Part may be taken as setting the goal which
inference must be able to reach, if our data and our principles
of inference are to justify scientific practice.

Part II, on language, is still concerned with preliminaries.
These are mainly of two sorts. On the one hand, it is important
to make clear the meanings of certain fundamental terms, such
as “fact” and “truth”. On the other hand, it is necessary to
examine the relation of sensible experience to empirical concepts
such as “red”, “hard”, “metre”, or “second”. In addition, we
shall examine the relation of words having an essential reference
to the speaker, such as “here” and “now”, to impersonal words,
such as those assigning latitude, longitude, and date. This raises
problems, of considerable importance and some difficulty, which
are concerned with the relation of individual experience to the
socially recognized body of general knowledge.

In Part III, on Science and Perception, we begin our main
inquiry. We are concerned, here, to disentangle data from
inferences in what ordinarily passes for empirical knowledge.

HUMAN knowledge: its scope and limits

We are not yet concerned to justify inferences, or to investigate
the principles according to which they are made, but we are
concerned to show that inference s (as opposed to logica l Xr ^
constructio ns) are necessary to science. We are concerned also
to distinguish between two kinds of space and time, one sub-
jective and appertaining to data, the other objective and inferred.
Incidentally we shall contend that solipsism, except in an extreme
form in which it has never been entertained, is an illogical half-
way house between the fragmentary world of data and the
complete world of science.

Part IV, on scientific concepts, is concerned to analyse the
fundamental concepts of the inferred scientific world, more
especially physical space, historical _time, and causal law s. The
terms employed in mathematical physics are required to fulfil
two kinds of conditions: on the one hand, they must satisfy
certain formulae; on the other hand, they must be so interpreted
as to yield results that can be confirmed or confuted by observa-
tion. Through the latter condition they are linked to data, though
somewhat loosely; through the former they become determinate
as regards certain structural properties. But considerable latitude
of interpretation remains. It is prudent to use this latitude in
such a way as to minimize the part played by inference as
opposed to construction; on this ground, for example, point-
instants in space-time are constructed as groups of events or of
qualities. Throughout this Part the two concepts of space-time
structure and causal chains assume a gradually increasing
importance.ftAs Part III was concerned to discover w hat can
be counted as data, so Part IV is concerned to set forth, in a
general wav, what, if science is to be j ustified, mimi-he afrle
to infer from our 5a tal

Since it is Admitted that scientific inferences, as a rule, only
confer probability on their conclusions, Part V proceeds to the
examination of Probability. This term is capable of various
interpretations, and has been differently defined by different
authors. These interpretations and definitions are examined, and
so are the attempts to connect induction with probability. In
this matter the conclusion reached is, in the main, that advocated
by Keynes: that inductions do not make their conclusions
probable unless certain conditions are fulfilled, and that experience
alone can never prove that these conditions are fulfilled.

INTRODUCTION

Part VI, on the postulates of scientific i nferen ce, endeavours
to discover what are the” minimum as sumptions, ante rior^to
experience, that ar e required to justify usTn inferring laws from
a^collection oFdat a; and further, to inqinre~~ih~ whaF sense, if
any, we can be said to know that These lis's^^

The main logical function tTiaf' t^^assumptiohs Tiave to fulfil
is that of conferring a high probability on the conclusions of
inductions that satisfy certain conditions. For this purpose,
since only probability is in question, we do not need to assume
that such-and-such a connection of events occurs always, but
only that it occurs frequently. For example, one of the assumptions
that appear necessary is that of separable causal chains, such as
are exhibited by light-rays or sound-waves. This assumption
can be stated as follows : when an event having a complex space-
time structure occurs, it frequently happens that it is one of a
train of events having the same or a very similar structure. (A
more exact statement will be found in Chapter VI of this Part.)
This is part of a wider assumption of regularity, or natural law,
which, however, requires to be stated in more specific forms than
is usual, for in its usual form it turns out to be a tautology.

That scientific inference requires, for its validity, principles
which experience cannot render even probable, is, I believe,
an inescapable conclusion from the logic of probability. For
empiricism, it is an awkward conclusion. But I think it can be
rendered somewhat more palatable by the analysis of the concept
of “knowledge” undertaken in Part II. “Knowledge”, in my
opinion, is a much less precise concept than is generally thought,
and has its roots more deeply embedded in unverbalized animal
behaviour than most philosophers have been willing to admit.
The logically basic assumptions to which our analysis leads us are
psychologically the end of a long series of refinements which
start from habits of expectation in animals, such as that what
has a certain kind of smell will be good to eat. To ask, therefore,
whether we “know” the postulates of scientific inference, is not
so definite a question as it seems. The answer must be: in one
sense, yes, in another sense, no; but in the sense in which “no”
is the right answer we know nothing whatever, and “knowledge”
in this sense is a delusive vision. The perplexities of philosophers
are due, in a large measure, to their unwillingness to awaken
from this blissful dream.

PART I

THE WORLD OF SCIENCE

Chapter I

INDIVIDUAL AND SOCIAL KNOWLEDGE

S cientific knowledge aims at being wholly impersonal, and
tries to state what has been discovered by the collective
intellect of mankind. In this chapter I shall consider how
far it succeeds in this aim, and what elements of individual
knowledge have to be sacrificed in order to achieve the measure
of success that is possible.

The community knows both more and less than the individual :
it knows, in its collective capacity, all the contents of the
Encyclopaedia and all the contributions to the Proceedings of
learned bodies, but it does not know the warm and intimate
things that make up the colour and texture of an individual life.
When a man says “I can never convey the horror I felt on seeing
Buchenwald” or “no words can express my joy at seeing the sea
again after years in a prison camp”, he is saying something which
is strictly and precisely true : he possesses, through his experience,
knowledge not possessed by those whose experience has been
different, and not completely capable of verbal expression. If he
is a superb literary artist he may create in sensitive readers a state
of mind not wholly unlike his own, but if he tries scientific methods
the stream of his experience will be lost and dissipated in a dusty
desert.

Language, our sole means of communicating scientific know-
ledge, is essentially social in its origin and in its main functions.
It is true that, if a mathematician were wrecked or a desert island
with a note-book and a pencil, he would, in all likelihood, seek to
make his solitude endurable by calculations using the language of
mathematics; it is true also that a man may keep a diary which
he intends to conceal from all eyes but his own. On a more every-
day plane, most of us use words in solitary thinking. Nevertheless
the chief purpose of language is communication, and to serve this
purpose it must be public, not a private dialect invented by the
speaker. It follows that what is most personal in each individual's
experience tends to evaporate during the process of ' translation
into language. What is more, the very publicity of language is in
large part a delusion. A given form of words will usually be

human knowledge: its scope and limits

interpreted by competent hearers in such a way as to be true for
all of them or false for all of them, but in spite of this it will not
have the same meaning for all of them. Differences which do not
affect the truth or falsehood of a statement are usually of little
practical importance, and are therefore ignored, with the result
that we all believe our private world to be much more like the
public world than it really is.

This is easily proved by considering the process of learning to
understand language. There are two ways of getting to know
what a word means : one is by a definition in terms of other words,
which is called verbal definition ; the other is by frequently hearing
the word when the object which it denotes is present, which is
called ostensive definition. It is obvious that ostensive definition is
alone possible in the beginning, since verbal definition pre-
supposes a knowledge of the words used in the definiens. You can
learn by a verbal definition that a pentagon is a plane figure with
five sides, but a child does not learn in this way the meaning of
every-day words such as “rain”, “sun”, “dinner”, or “bed”.
These are taught by using the appropriate word emphatically
while the child is noticing the object concerned. Consequently the
meaning that the child comes to attach to the word is a product of
his personal experience, and varies according to his circumstances
and his sensorium. A child who frequently experiences a mild
drizzle will attach a different idea to the word “rain” from that
formed by a child who has only experienced tropical torrents. A
short-sighted and a long-sighted child will connect different
images with the word “bed”.

It is true that education tries to depersonalize language, and
with a certain measure of success. “Rain” is no longer the familiar
phenomenon, but “drops of water falling from clouds towards the
earth”, and “water” is no longer what makes you wet, but H 2 0.
As for hydrogen and oxygen, they have verbal definitions which
have to be learnt by heart ; whether you understand them does not
matter. And so, as your instruction proceeds, the world of words
becomes more and more separated from the world of the senses ;
you acquire the art of using words correctly, as you might acquire
the art of playing the fiddle; in the end you become such a
virtuoso in the manipulation of phrases that you need hardly ever
remember that words have meanings. You have then become
completely a public character, and even your inmost thoughts are

INDIVIDUAL AND SOCIAL KNOWLEDGE

suitable for the encyclopaedia. But you can no longer hope to be
a poet, and if you try to be a lover you will find your depersonalized
language not very successful in generating the desired emotions.
You have sacrificed expression to communication, and what you
can communicate turns out to be abstract and dry.

It is an important fact that the nearer we come to the complete
abstractness of logic, the less is the unavoidable difference between
different people in the meaning attached to a word. I see no
reason why there should be any difference at all between two
suitably educated persons in the idea conveyed to them by the
word “3481”. The words “or” and “not” are capable of having
exactly the same meaning for two different logicians. Pure
mathematics, throughout, works with concepts which are capable
of being completely public and impersonal. The reason is that
they derive nothing from the senses, and that the senses are the
source of privacy. The body is a sensitive recording instrument,
constantly transmitting messages from the outside world; the
messages reaching one body are never quite the same as those
reaching another, though practical and social exigencies have
taught us ways of disregarding the differences between the
percepts of neighbouring persons. In constructing physics we
have emphasized the spatio-temporal aspect of our perceptions,
which is the aspect that is most abstract and most nearly akin to
logic and mathematics. This we have done in the pursuit of
publicity, in order to communicate what is communicable and to
cover up the rest in a dark mantle of oblivion.

Space and time, however, as human beings know them, are not
in reality so impersonal as science pretends. Theologians conceive
God as viewing both space and time from without, impartially,
and with a uniform awareness of the whole; science tries to
imitate this impartiality with some apparent success, but the
success is in part illusory. Human beings differ from the
theologians* God in the fact that their space and time have a
here and now. What is here and now is vivid, what is remote has
a gradually increasing dimness. All our knowledge of events
radiates from a space-time centre, which is the little region that
we are occupying at the moment. “Here” is a vague term: in
astronomical cosmology the Milky Way may count as “here”, in
the study of the Milky Way “here” is the solar system, in the
study of the solar system “here” is the earth, in geography it is

human knowledge: its scope and limits

the town or district in which we live, in physiological studies of
sensation it is the brain as opposed to the rest of the body. Larger
“heres” always contain smaller ones as parts; all “heres” contain
the brain of the speaker, or part of it. Similar considerations apply
to “now”.

Science professes to eliminate “here” and “now”. When some
event occurs on the earth's surface, we give its position in the
space-time manifold by assigning latitude, longitude, and date.
We have developed a technique which insures that all accurate
observers with accurate instruments will arrive at the same
estimate of latitude, longitude, and date. Consequently there is no
longer anything personal in these estimates, in so far as we are
content with numerical statements of which the meaning is not
too closely investigated. Having arbitrarily decided that the
longitude of Greenwich and the latitude of the equator are to
be zero, other latitudes and longitudes follow. But what is
“Greenwich”? This is hardly the sort of term that ought to
occur in an impartial survey of the universe, and its definition is
not mathematical. The best way to define “Greenwich” is to take
a man to it and say: “Here is Greenwich.” If some one else has
already determined the latitude and longitude of the place where
you are, “Greenwich” can be defined by its latitude and longitude
relative to that place; it is, for example, so many degrees east
and so many degrees north of New York. But this does not get
rid of “here”, which is now New York instead of Greenwich.

Moreover it is absurd to define either Greenwich or New York
by its latitude and longitude. Greenwich is an actual place,
inhabited by actual people, and containing buildings which ante-
date its longitudinal pre-eminence. You can, of course, describe
Greenwich, but there always might be another town with the
same characteristics. If you want to be sure that your description
applies to no other place, the only way is to mention its relation
to some other place, for instance, by saying that it is so many
miles down the Thames from London Bridge. But then you will
have to define “London Bridge”. Sooner or later you are faced
with the necessity of defining some place as “here”, and this is
an egocentric definition, since the place in question is not “here”
for everybody. There may be a way of escape from this con-
clusion; at a later stage, we will resume the question. But there
is no obvious or easy way of escape, and until one is found all

INDIVIDUAL AND SOCIAL KNOWLEDGE

determinations of latitude and longitude are infected with the
subjectivity of “here”. This means that, although different people
assign the same latitude and longitude to a place, they do not, in
ultimate analysis, attach the same meaning to the figures at which
they arrive.

The common world in which we believe ourselves to live is a
construction, partly scientific, partly pre-scientific. We perceive
tables as circular or rectangular, in spite of the fact that a painter,
to reproduce their appearance, has to paint ellipses or non-
rectangular quadrilaterals. We see a person as of about the same
size whether he is two feet from us or twelve. Until our attention
is drawn to the facts, we are quite unconscious of the corrections
that experience has led us to make in interpreting sensible appear-
ances. There is a long journey from the child who draws two eyes
in a profile to the physicist who talks of electrons and protons,
but throughout this journey there is one constant purpose: to
eliminate the subjectivity of sensation, and substitute a kind of
knowledge which can be the same for all percipients. Gradually
the difference between what is sensed and what is believed to be
objective grows greater; the child’s profile with two eyes is still
very like what is seen, but the electrons and protons have only a
remote resemblance of logical structure. The electrons and
protons, however, have the merit that they may be what actually
exists where there are no sense-organs, whereas our immediate
visual data, owing to their subjectivity, are almost certainly not
what takes place in the physical objects that we are said to see.

The electrons and protons — assuming it scientifically correct to
believe in them — do not depend for their existence upon being
perceived; on the contrary, there is every reason to believe that
they existed for countless ages before there were any percipients
in the universe. But although perception is not needed for their
existence, it is needed to give us a reason for believing in their
existence. Hundreds of thousands of years ago, a vast and remote
region emitted incredible numbers of photons, which wandered
through the universe in all directions. At last a very few of them
hit a photographic plate, in which they caused chemical changes
which made parts of the plate look black instead of whjte when
examined by an astronomer. This tiny effect upon a minute but
highly educated organism is our only reason for believing in the
existence of a nebula comparable in size with the Milky Way.

human knowledge: its scope and limits

ijfer The order for knowled ge is the inverse of the causal order. In the
order" for knowledge, what comes first is the brief subjective
experience of the astronomer looking at a pattern of black and
white, and what comes last is the nebula, vast, remote, and
belonging to the distant past.

In considering the reasons for believing in any empirical
statement, we cannot escape from perception with all its personal
limitations. How far the information which we obtain from this
tainted source can be purified in the filter of scientific method,
and emerge resplendently godlike in its impartiality, is a difficult
question, with which we shall be much concerned. But there is
one thing that is obvious from the start: only in so far as the
initial perceptual datum is trustworthy can there be any reason
for accepting the vast cosmic edifice of inference which is based
upon it.

I am not suggesting that the initial perceptual datum must be
accepted as indubitable ; that is by no means the case. There are
well-known methods of strengthening or weakening the force of
individual testimony; certain methods are used in the law courts,
somewhat different ones are used in science. But all depend upon
the principle that some weight is to be attached to every piece of
testimony, for it is only in virtue of this principle that a number
of concordant testimonies are held to give a high probability.
Individual percepts are the basis of all our knowledge, and no
method exists by which we can begin with data which are public
to many observers.

Chapter II

THE UNIVERSE OF ASTRONOMY

Jk stronomy is the oldest of the sciences, and the contempla-

L\ tion of the heavens, with their periodic regularities, gave
A. Vmen their first conceptions of natural law. But in spite of
its age, astronomy is as vigorous as at any former time, and as
important in helping us to form a just estimate of man’s position
in the universe.

When the Greeks began inventing astronomical hypotheses, the
apparent motions of the sun and moon and planets among the
fixed stars had already been observed for thousands of years by
the Babylonians and Egyptians, who had also learned to predict
lunar eclipses with certainty and solar eclipses with a considerable
risk of error. The Greeks, like other ancient nations, believed the
heavenly bodies to be gods, or at any rate each closely controlled
by its own god or goddess. Some, it is true, questioned this opinion :
Anaxagoras, in the time of Pericles, maintained that the sun was
a red-hot stone and that the moon was made of earth. But for this
opinion he was prosecuted and compelled to fly from Athens. It
is very questionable whether either Plato or Aristotle was equally
rationalistic. But it was not the most rationalistic among the
Greeks who were the best astronomers; it was the Pythagoreans,
to whom superstition suggested what happened to be good
hypotheses.

The Pythagoreans, towards the end of the fifth century b.c.,
discovered that the earth is spherical; about a hundred years
later, Eratosthenes estimated the earth’s diameter correctly within
about fifty miles. Heraclides of Pontus, during the fourth century,
maintained that the earth rotates once a day and that Venus and
Mercury describe orbits about the sun. Aristarchus of Samos, in
the third century, advocated the complete Copernican system,
and worked out a theoretically correct method of estimating the
distances of the sun and moon. As regards the sun this result, it
is true, was wildly wrong, owing to inaccuracy in his data; but a
hundred years later Posidonius made an estimate which was
about half of the correct figure. This extraordinarily vigorous
advance, however, did not continue, and much of it was for-

human knowledge: its scope and limits

gotten in the general decay of intellectual energy during later
antiquity.

The cosmos, as it appears, for instance, in Plotinus, was a cosy
and human little abode in comparison with what it has since
become. The supreme deity regulated the whole, but each star
was a subordinate deity, similar to a human being but in every
way nobler and wiser. Plotinus finds fault with the Gnostics for
believing that, in the created universe, there is nothing more
worthy of admiration than the human soul. The beauty of the
heavens, to him, is not only visual, but also moral and intellectual.
The sun and moon and planets are exalted spirits, actuated by
such motives as appeal to the philosopher in his best moments.
He rejects with indignation the morose view of the Gnostics (and
later of the Manicheans) that the visible world was created by a
wicked Demiurge and must be despised by every aspirant to true
virtue. On the contrary, the bright beings that adorn the sky are
wise and good, and such as to console the philosopher amid the
welter of folly and disaster that was overtaking the Roman Empire.

The medieval Christian cosmos, though less austere than that
of the Manicheans, was shorn of some elements of poetic fancy
that paganism had preserved to the end. The change, however,
was not very great, for angels and archangels more or less took
the place of the polytheists’ celestial divinities. Both the scientific
and the poetic elements of the medieval cosmos are set forth in
Dante’s Paradiso ; the scientific elements are derived from
Aristotle and Ptolemy. The earth is spherical, and at the centre
of the universe ; Satan is at the centre of the earth, and hell is an
inverted cone of which he forms the apex. At the antipodes of
Jerusalem is the Mount of Purgatory, at whose summit is the
earthly paradise, which is just in contact with the sphere of the
moon.

The heavens consist of ten concentric spheres, that of the moon
being the lowest. Everything below the moon is subject to cor-
ruption and decay; everything from the moon upwards is in-
destructible. Above the moon, the spheres in their order are those
of Mercury, Venus, the Sun, Mars, Jupiter, Saturn and the fixed
stars, beyond which is the Primum Mobile. Last of all, above the
Primum Mobile, is the Empyrean, which has no motion, and in
which there are no times or places. God, the Aristotelian Unmoved
Mover, causes the rotation of the Primum Mobile, which, in turn,

THE UNIVERSE OF ASTRONOMY

communicates its motion to the sphere of the fixed stars, and so
on downwards to the sphere of the moon. Nothing is said in
Dante as to the sizes of the various spheres, but he is able to
traverse them all in the space of twenty-four hours. Clearly the
universe as he conceived it was somewhat minute by modern
standards; it was also very recent, having been created a few
thousand years ago. The spheres, which all had the earth at the
centre, afforded the eternal abodes of the elect. The elect consisted
of those baptized persons who had reached the required standard
both in faith and works, together with the patriarchs and prophets
who had foreseen the coming of Christ, and a very few pagans
who, while on earth, had been miraculously enlightened.

It was against this picture of the universe that the pioneers of
modern astronomy had to contend. It is interesting to contrast the
commotion about Copernicus with the almost complete oblivion
that befell Aristarchus. Cleanthes the Stoic had urged that
Aristarchus should be prosecuted for impiety, but the Govern-
ment was apathetic; perhaps if he had been persecuted, like
Galileo, his theories might have won wider publicity. There were,
however, other more important reasons for the difference between
the posthumous fame of Aristarchus and that of Copernicus. In
Greek times astronomy was an amusement of the idle rich — a very
dignified amusement, it is true, but not an integrated part of the
life of the community. By the sixteenth century, science had
invented gunpowder and the mariner’s compass, the discovery of
America had shown the limitations of ancient geognosis, Catholic
orthodoxy had begun to seem an obstacle to material progress, and
the fury of obscurantist theologians made the men of science
appear as heroic champions of a new wisdom. The seventeenth
century, with the telescope, the science of dynamics, and the law
of gravitation, completed the triumph of the scientific outlook,
not only as the key to pure knowledge, but as a powerful means
of economic progress. From this time onwards, science was
recognized as a matter of social and not merely individual interest.

The theory of the sun and planets as a finished system was
practically completed by Newton. As against Aristotle and the
medieval philosophers it appeared that the sun, not the earth, is
the centre of the solar system; that the heavenly Bodies, left to
themselves, would move in straight lines, not in circles; that in
fact they move neither in straight lines nor in circles, but in

2 5

human knowledge: its scope and limits

ellipses ; and that no action from outside is necessary to preserve
their motion. But as regards the origin of the system Newton had
nothing scientific to say; he supposed that at the Creation the
planets had been hurled by the hand of God in a tangential
direction, and had then been left by Him to the operation of the
law of gravitation. Before Newton, Descartes had attempted a
theory of the origin of the solar system, but his theory proved
untenable. Kant and Laplace invented the nebular hypothesis,
according to which the sun was formed by the condensation of a
primitive nebula, and threw off the planets successively as a
result of increasingly rapid rotation. This theory also proved
defective, and modern astronomers incline to the view that the
planets were caused by the passage of another star through the
near neighbourhood of the sun. The subject remains obscure, but
no one doubts that, by some mechanism, the planets came out of
the sun.

The most remarkable astronomical progress in recent times has
been in relation to the stars and the nebulae. The nearest of the
fixed stars, Alpha Centauri, is at a distance of about 25 X io 12
miles, or 4-2 light-years. (Light travels 186,000 miles a second;
a light-year is the distance it travels in a year.) The first deter-
mination of the distance of a star was in 1835; since then, by
various ingenious methods, greater and greater distances have been
computed. It is believed that the most distant object that can be
detected with the most powerful telescope now in existence is
about 500 million light-years away.

Something is now known of the general structure of the
universe. The sun is a star in the galaxy, which is an assembly of
about 300,000 million stars, about 150,000 light-years across and
between 25,000 and 40,000 light-years thick. The total mass of
the galaxy is about 160,000 million times the mass of the sun;
the mass of the sun is about 2 X io 27 tons. The whole of this
system is slowly rotating about its centre of gravity ; the sun takes
about 225 million years to complete its orbit round the milky way.

In the space beyond the milky way, other systems of stars, of
approximately the same size as the milky way, are scattered at
fairly regular intervals throughout the space that our telescopes
can explore. These systems are called extra-galactic nebulae ; it is
thought that about 30 millions of them are visible, but the census
is not yet complete. The average distance between two nebulae is

THE UNIVERSE OF ASTRONOMY

about 2 million light-years. (Most of these facts are taken from
Hubble, The Realm of the Nebulae , 1936.)

One of the oddest facts about the nebulae is that the lines in
their spectra, with very few exceptions, are shifted towards the
red, and that the amount of the shift is proportional to the distance
of the nebula. The only plausible explanation is that the nebulae
are moving away from us, and that the most distant ones are
receding most quickly. At a distance of 135 million light-years,
this velocity amounts to 14,300 miles per second (Hubble, Plate
VIII, p. 1 18). At a certain distance, the velocity would become
equal to the velocity of light, and the nebulae would therefore be
invisible however powerful our telescopes might be.

The general theory of relativity has an explanation to offer of
this curious phenomenon. The theory maintains that the universe
is of finite size — not that it has an edge, outside which there is
something which is not part of the universe, but that it is a three-
dimensional sphere, in which the straightest possible lines return
in time to their starting-point, as on the surface of the earth. The
theory goes on to predict that the universe must be either con-
tracting or expanding; it then uses the observed facts about the
nebulae to decide for expansion. According to Eddington, the
universe doubles in size every 1,300 million years or so. (New
Pathways in Science , p. 210.) If this is true, the universe was once
quite small, but will in time become rather large.

This brings us to the question of the ages of the earth and the
stars and the nebulae. On grounds that are largely geological, the
age of the earth is estimated at about 3,000 million years. The age
of the sun and the other stars is still a matter of controversy. If,
in the interior of a star, matter can be annihilated by transforming
an electron and a proton into radiation, the stars may be several
million million years old; if not, only a few thousand million.
(H. Spencer Jones, Worlds Without End , p. 231.) On the whole,
the latter view seems to be prevailing.

There is even some reason to think that the universe had a
beginning in time; Eddington used to maintain that it began in
about 90,000 million B.c, This is certainly more than the 4,004
in which our great-grandfathers believed, but it is still a finite
period, and raises all the old puzzles as to what was going on
before that date.

The net result of this summary survey of the astronomical

HUMAN knowledge: its scope and limits

When the forces to which a body is subject are not constant, the
principle does not allow us to take each separately for a finite
time, but if the finite time is short the result of taking each
separately will be approximately right, and the shorter the time
the more nearly right it will be, approaching complete rightness
as a limit.

It must be understood that this law is purely empirical ; there
is no mathematical reason for its truth. It is to be believed in so
far as there is evidence for it, and no further. In quantum
mechanics it is not assumed, and there are phenomena which
seem to show that it is not true in atomic occurrences. But in the
physics of large-scale occurrences it remains true, and in classical
physics it played a very important role.

From Newton to the end of the nineteenth century, the progress
of physics involved no basically new principles. The first revo-
lutionary novelty was Planck's introduction of the quantum
constant h in the year 1900. But before considering quantum
theory, which is chiefly important in connection with the structure
and behaviour of atoms, a few words must be said about
relativity, which involved a departure from Newtonian principles
much slighter than that of quantum theory.

^Newton believed that, in addition to matter, there is absolute
space and absolute time. That is to say, there is a three- dimensional
manifold of points and a one-dimensional manifold of instant s,
and there is~^TRree-term relation involving matter* space, and
timejliamely the relation of “occupying” a point at an instant.
In this view Newton agreed with Democritus and the other
atomists of antiquity, who believed in “atoms and the void".
Other philosophers had maintained that empty space is nothing,
and that there must be matter everywhere. This was Descartes'
opinion, and also that of Leibniz, with whom Newton (using Dr.
Clarke as his mouthpiece) had a controversy on the subject.

Whatever physicists might hold as a matter of philosophy,
Newton’s view was implicit in the technique of dynamics, and
there were, as he pointed out, empirical reasons for preferring it.
If water in a bucket is rotated, it climbs up the sides, but if the
bucket is rotated while the water is kept still, the surface of the
water remains flat. We can therefore distinguish between rotation
of the water and rotation of the bucket, which we ought not to be
able to do if rotation were merely relative. Since Newton's time

3 *

THE WORLD OF PHYSICS

other arguments of the same sort have accumulated. Foucault’s
pendulum, the flattening of the earth at the poles, and the fact
that bodies weigh less in low latitudes than in high ones, would
enable us to infer that the earth rotates even if the sky were always
covered with clouds; in fact, on Newtonian principles we can say
that the rotation of the earth, not the revolution of the heavens,
causes the succession of night and day and the rising and setting
of the stars. But if space is purely relative, the difference between
the statements “the earth rotates” and “the heavens revolve” is
purely verbal : both must be ways of describing the same pheno-
mena.

^Einstein showed how to avoid Newton’s conclusions, and make
spatio-temporal position purely relative. But his theory of
relativity did much more than this. In the special theory of
relativity he showed that between two events there is a relation,
which may be called “interval”, which can be divided in many
different ways into what we should regard as a spatial distance
and what we should regard as a lapse of time. All these different
ways are equally legitimate ; there is not one way which is more
“right” than the others. The choice between them is a matter of
pure convention, like the choice between the metric system and
the system of feet and inches !j

It follows from this that the fundamental manifold of physics
cannot consist of persistent particles in motion, but must consist
of a four-dimensional manifold of “events’*? There will be three
co-ordinates to fix the position of the event in space, and one to
fix its position in time, but a change of co-ordinates may alter the
time-co-ordinate as well as the space co-ordinates, and not only,
as before, by a constant amount, the same for all events — as, for
example, when dating is altered from the Mohammedan era to
the Christian.

The general theory of relativity — published in 1915, ten years
after the special theory — was primarily a geometrical theory of
gravitation. This part of the theory may be considered firmly
established. But it has also more speculative features. It contains,
in its equations, what is called the “cosmical constant”, which
determines the size of the universe at any time. This part of the
theory, as I mentioned before, is held to show that the universe
is growing either continually larger or continually smaller.^ The
shift towards the red in the spectra of distant nebulae is held to

human knowledge: its scope and limits

show that they are moving away from us with a velocity pro-
portional to their distance from us. This leads to the conclusion
that the universe is expanding, not contracting. It must be under-
stood that, according to this theory, the universe is finite but
unbounded, like the surface of a sphere, but in three dimensions.
All this involves non-Euclidean geometry, and is apt to seem
mysterious to those whose imagination is obstinately Euclidean.

Two kinds of departure from Euclidean space are involved in
the general theory of relativity. On the one hand, there are what
may be called the small-scale departures (where the solar system,
e.g., is regarded as “small”), and on the other hand the large-scale
departure of the universe as a whole. The small-scale departures
occur in the neighbourhood of matter, and account for gravitation.
They may be compared to hills and valleys on the surface of the
earth. The large-scale departure may be compared with the fact
that the earth is round and not flat. If you start from any point
on the earth’s surface and travel as straight as you can, you will
ultimately return to your starting-point. So, it is held, the
straightest line possible in the universe will ultimately return into
itself. The analogy with the surface of the earth fails in that the
earth’s surface is two-dimensional and has regions outside it,
whereas the spherical space of the universe is three-dimensional
and has nothing outside it. The present circumference of the
universe is between 6,000 and 60,000 million light-years, but the
size of the universe is doubled about every 1,300 million years.
All this, however, must still be regarded as open to doubt.

According to Professor E. A. Milne, 1 there is a great deal more
that is questionable in Einstein’s theory. Professor Milne holds
that there is no need to regard space as non-Euclidean, and that
the geometry we adopt can be decided entirely by motives of
convenience. The difference between different geometries, accord-
ing to him, is a difference in language, not in what is described.
Where physicists disagree it is rash for an outsider to have an
opinion, but I incline to think that Professor Milne is very likely
to be in the right.

^. Quantum theory , i n co ntrast to the theory of relativity, is
mainly concerned with the smallest things about which knowledge
is pdssiBTe, Tiamely atoms and their structure. During the nine-

1 Relativity Gravitation qnd World Structure . By E. A. Milne. Oxford,

* 935 -

THE WORLD OF PHYSICS

teenth century the atomic constitution of matter became well
established, and it was found that the different elements could be
placed in a series starting with hydrogen and ending with uranium.
The place of an element in this series is called its “atomic
number”. Hydrogen has the atomic number i, and uranium 92.
There are two gaps in the series at present, so that the number of
known elements is 90, not 92; but the gaps may be filled any day,
as a number of previously existing gaps have been. In general, but
not always, the atomic number increases with the atomic weight.
Before Rutherford, there was no plausible theory as to the structure
of atoms, or as to the physical properties which caused them to
fall into a series. The series was determined by their chemical
properties alone, and of these properties no physical explanation
existed.

The Rutherford-Bohr atom, as it is called after its two inventors,
had a beautiful simplicity, now, alas, lost. But although it has
become only a pictorial approximation to the truth, it can still be
used when extreme accuracy is not required, and without it the
modern quantum theory could never have arisen. It is therefore
still necessary to say something about it.

Rutherford gave experimental reasons for regarding an atom as
composed of a nucleus carrying positive electricity surrounded by
very much lighter bodies, called “electrons”, which carried
negative electricity, and revolved, like planets, in orbits about the
nucleus. When the atom is not electrified, the number of planetary
electrons is the atomic number of the element concerned ; at all times,
the atomic number measures the net positive electricity carried
by the nucleus. The hydrogen atom consists of a nucleus and one
planetary electron; the nucleus of the hydrogen atom is called a
“proton”. It was found that the nuclei of other elements could be
regarded as composed of protons and electrons, the number of
protons being greater than that of the electrons by the atomic
number of the element. Thus helium, which is number 2, has a
nucleus consisting of four protons and two electrons. The atomic
weight is practically determined by the number of protons, since
a proton has about 1,850 times the mass of an electron, so that the
contribution of the electrons to the total mass is almost negligible.

It has been found that, in addition to electrons and protons,
there are two other constituents of atoms, which are called
“positrons” and “neutrons”. A positron is just like an electron,

HUMAN knowledge: its scope and limits

except that it carries positive instead of negative electricity; it has
the same mass as an electron, and probably the same size, in so
far as either can be said to have a size. The neutron has no
electricity, but has approximately the same mass as a proton. It
seems not unlikely that a proton consists of a positron and a
neutron. If so, there are three ultimate kinds of constituents in
the perfected Rutherford-Bohr atom: the neutron, which has
mass but no electricity, the positron, carrying positive electricity,
and the electron, carrying an equal amount of negative electricity.

But we must now return to theories which ante-date the
discovery of neutrons and positrons.

Bohr added to the Rutherford picture a theory as to the possible
orbits of electrons, which, for the first time, explained the lines
in the spectrum of an element. This mathematical explanation was
almost, but not quite, perfect in the cases of hydrogen and
positively electrified helium; in other cases the mathematics was
too difficult, but no reason appeared to suppose that the theory
would give wrong results if the mathematics could be worked out.
His theory made use of Planck’s quantum constant A, concerning
which a few words must be said.

Planck, by studying radiation, proved that in a light or heat
wave of frequency v the energy must be h.v or 2 h.v or 3 h.v or
some other integral multiple of A . v , where A is “Planck’s con-
stant”, of which the value in C.G.S. units is about 6-55 x 10“ 27 ,
and the dimensions are those of action, i.e. energy x time. Before
Planck, it had been supposed that the energy of a wave could vary
continuously, but he showed conclusively that this could not be
the case. The frequency of waves is the number that pass a given
point in a second. In the case of light, the frequency determines
the colour; violet light has the highest frequency, red light the
lowest. There are other waves of just the same kind as light- waves,
but not having the frequencies that cause visual sensations of
colour. Higher frequencies than those of violet light are, in order,
ultra-violet, X-rays and y-rays; lower frequencies, infra-red and
those used in wireless telegraphy.

When an atom emits light, it does so because it has parted with
an amount of energy equal to that in the light-wave. If it emits
light of frequency v , it must, according to Planck’s theory, have
parted with an amount, of energy measured by A . v or some
integral multiple of h.v. Bohr supposed that this happened through

THE WORLD OF PHYSICS

a planetary electron jumping from a larger to a smaller orbit;
consequently the change of orbit must be such as to involve a loss
of energy h . v or some integral multiple of this amount. It followed
that only certain orbits could be possible. In the hydrogen atom,
there would be a smallest possible orbit, and the other possible
ones would have 4, 9, 16, . . . times the radius of the minimum
orbit. This theory, first propounded in 1913, was found to agree
well with observation, and for a time won general acceptance.
Gradually, however, it was found that there were facts which it
could not explain, so that, though clearly a step on the way to the
truth, it could no longer be accepted as it stood. The new and
more radical quantum theory, which dates from 1925, is due in
the main to two men, Heisenberg and Schrodinger.

In the modern theory there is no longer any attempt to make
an imaginative picture of the atom. An atom only gives evidence
of its existence when it emits energy, and therefore experimental
evidence can only be of changes of energy. The new theory takes
over from Bohr the doctrine that the energy in an atom must have
one of a discrete series of values involving h; each of these is
called an “energy lever’. But as to what gives the atom its energy
the theory is prudently silent.

One of the oddest things about the theory is that it has abolished
the distinction between waves and particles. Newton thought that
light consisted of particles emitted by the source of the light;
Huygens thought that it consisted of waves. The view of Huygens
prevailed, and until recently was thought to be definitely estab-
lished. But new experimental facts seemed to demand that light
should consist of particles, which were called “photons”. Per
contra, De Broglie suggested that matter consists of waves. In
the end it was shown that everything in physics can be explained
either on the particle hypothesis or on the wave hypothesis.
There is therefore no physical difference between them, and
either may be adopted in any problem as may suit our convenience.
But whichever is adopted, it must be adhered to; we must not
mix the two hypotheses in one calculation.

In quantum theory, individual atomic occurrences are not
determined by the equations ; these suffice only to show that the
possibilities form a discrete series, and that there' are rules
determining how often each possibility will be realized in a large
number of c*ses. There are reasons for believing that this absence

HUMAN knowledge: its scope and limits

of complete determinism is not due to any incompleteness in the
theory, but is a genuine characteristic of small-scale occurrences.
The regularity which is found in macroscopic phenomena is a
statistical regularity. Phenomena involving large numbers of
atoms remain deterministic, but what an individual atom may do
in given circumstances is uncertain, not only because our
knowledge is limited, but because there are no physical laws
giving a determinate result.

There is another result of quantum theory, about which, in
my opinion, too much fuss has been made, namely what is called
Heisenberg’s uncertainty-principle. According to this there is a
theoretical limit to the accuracy with which certain connected
quantities can be simultaneously measured. In specifying the state
of a physical system, there are certain pairs of connected quantities ;
one such pair is position and momentum (or velocity, so long
as the mass is constant), another is energy and time. It is of course
a commonplace that no physical quantity can be measured with
complete accuracy, but it had always been supposed that there
was no theoretical limit to the increase of accuracy obtainable
by improved technique. According to Heisenberg’s principle this
is not the case. If we try to measure simultaneously two con-
nected quantities of the above sort, any increase of accuracy in
the measurement of one of them (beyond a certain point) involves
a decrease in the accuracy of the measurement of the other. In
fact, there will be errors in both measurements, and the product
of these two errors can never be less than hjzTT. This means that,
if one could be completely accurate, the error in the other would
have to be infinite. Suppose, for instance, that you wish to deter-
mine the position and velocity of a particle at a certain time: if
you get the position very nearly right, there will be a large error
in the velocity, and if you get the velocity very nearly right, there
will be a large error as to the position. Similarly as regards energy
and time: if you measure the energy very accurately, the time
when the system has this energy will have a large margin of un-
certainty, while if you fix the time very accurately the energy will
become uncertain within wide limits. This is not a question of
imperfection in our measuring apparatus, but is an essential
principle of physics.

There are physical considerations which make this principle
less surprising. It will be observed that h is a very small quantity,

THE WORLD OF PHYSICS

since it is of the order of io" 27 . Therefore wherever h is relevant
we are concerned with matters involving very great minuteness.
When an astronomer observes the sun, the sun preserves a lordly
indifference to his proceedings. But when a physicist tries to find
out what is happening to an atom, the apparatus by means of
which he makes his observations is likely to have an effect upon
the atom. Detailed considerations show that the sort of apparatus
best suited for determining the position of an atom is likely to
affect its velocity, while the sort of apparatus best suited for
determining its velocity is likely to alter its position. Similar
arguments apply to other pairs of related quantities. I do not
think, therefore, that the uncertainty principle has the kind of
philosophical importance that is sometimes attributed to it.

Quantum equations differ from those of classical physics in
a very important respect, namely that they are not “linear”. This
means that when you have discovered the effect of one cause
alone, and then the effect of another cause alone, you cannot find
the effect of both together by adding the two previous effects. This
has very odd results. Suppose, for instance, that you have a screen
with a small slit, and you bombard it with particles; some of these
will get through the slit. Suppose now you close the first slit and
make a second ; then some will get through the second slit. Now
open both slits at once. You would think that the number getting
through both slits would be the sum of the previous numbers,
but this turns out not to be the case. The behaviour of the par-
ticles at one slit seems to be affected by the existence of the other
slit. The equations are such as to predict this result, but it remains
surprising. In quantum mechanics there is less independence of
causes than in classical physics, and this adds greatly to the
difficulty of the calculations.

^Both relativity and quantum theory Jiave had the effect of
replacing" the olef conception of “mass” by that of “energy”.
“Mass” used to be defined as “quantity of matter”; “matter”
was, on the one hand “substance” in the metaphysical sense, and
on the other hand the technical form of the common-sense
notion of “thing”. “Energy” was, in its early stages, a state of
“matter”. It consisted of two parts, kinetic and potential. The
kinetic energy of a particle is half the product of the' mass and
the square of the velocity. The potential energy is measured by
the work that would have to be done to bring the particle to its

HUMAN knowledge: its scope and limits

present position from some standard position. (This leaves a
constant undetermined, but that is of no consequence.) If you
carry a stone from the ground to the top of a tower, it acquires
potential energy in the process; if you drop it from the top, the
potential energy is gradually transformed into kinetic energy
during the fall. In any self-contained system, the total energy is
constant. There are various forms of energy, of which heat is one;
there is a tendency for more and more of the energy in the universe
to take the form of heat. The conservation of energy first became
a well-grounded scientific generalization when Joule measured
the mechanical equivalent of heat.

Relativity theory and experiment both showed that mass is not
constant, as had been held, but is increased by rapid motion;
if a particle could move as fast as light, its mass would become
infinite. Since all motion is relative, the different estimates of mass
formed by different observers, according to their motion relative
to the particle in question, are all equally legitimate. So far as this
theory is concerned, however, there is still one estimate of mass
which may be considered fundamental, namely the estimate made
by an observer who is at rest relatively to the body whose mass
is to be measured. Since the increase of mass with velocity is only
appreciable for velocities comparable with that of light, this case
covers practically all observations except those of a and j8 particles
ejected from radio-active bodies.

Quantum theory has made a greater inroad upon the concept
of “mass”. It now appears that whenever energy is lost by radia-
tion there is a corresponding loss of mass. The sun is held to
be losing mass at the rate of four million tons a second. To take
another instance: a helium atom, unelectrified, consists (in the
language of Bohr’s theory) of four protons and four electrons,
while a hydrogen atom consists of one proton and one electron.
It might have been supposed that, assuming this to be the case,
the mass of a helium atom would be four times that of a hydrogen
atom. This, however, is not the case: taking the mass of the
helium atom as 4, that of the hydrogen atom is not 1, but 1 -008.
The reason is that energy is lost (by radiation) when four hydrogen
atoms combine to form one helium atom — at least so we must
suppose, for the process is not one which has ever been
observed.

It is thought that the combination of four hydrogen atoms to

THE WORLD OF PHYSICS

form one atom of helium occurs in the interior of stars, and could
be made to occur in terrestrial laboratories if we could produce
temperatures comparable to those in the interior of stars. Almost
all the loss of energy involved in building up elements other than
hydrogen occurs in the transition to helium; in later stages the
loss of energy is small. If helium, or any element other than
hydrogen, could be artificially manufactured out of hydrogen,
there would be in the process an enormous liberation of energy
in the form of light and heat. This suggests the possibility of
atomic bombs more destructive than the present ones, which are
made by means of uranium. There would be a further advantage:
the supply of uranium in the planet is very limited, and it is
feared that it may be used up before the human race is exter-
minated, but if the practically unlimited supply of hydrogen in
the sea could be utilized there would be considerable reason to
hope that homo sapiens might put an end to himself, to the great
advantage of the other less ferocious animals.

But it is time to return to less cheerful topics.

The language of Bohr’s theory is still adequate for many pur-
poses, but not for stating the fundamental principles of quantum
physics. To state these principles, we must avoid all pictures of
what goes on in an atom, and must abandon attempts to say what
energy is. We must say simply: there is something quantitative,
to which we give the name “energy”; this something is very
unevenly distributed in space; there are small regions in which
there is a great deal of it, which are called “atoms”, and are those
in which, according to older conceptions, there was matter; these
regions are perpetually absorbing or emitting energy in forms
that have a periodic “frequency”. Quantum equations give rules
determining the possible forms of energy emitted by a given atom,
and the proportion of cases (out of a large number) in which each
of the possibilities will be realized. Everything here is abstract
and mathematical except the sensations of colour, heat, etc.,
produced by the radiant energy in the observing physicist.

Mathematical physics contains such an immense superstructure
of theory that its basis in observation tends to be obscured. It
is, however, an empirical study, and its empirical character appears
most unequivocally where the physical constants are concerned.
Eddington {New Pathways in Science , p. 230) gives the following
list of the primitive constants of physics :

HUMAN knowledge: its scope and limits

very different from the temperatures to which we are accustomed ;
at these temperatures, molecules of a very high degree of com-
plexity can come into existence.

What distinguishes living from dead matter? Primarily, its
chemical constitution and cell structure. It is to be supposed that
its other characteristics follow from these. The most notable of
these others are assimilation and reproduction, which, in the
lowest forms of life, are not very sharply distinguished from each
other. The result of assimilation and reproduction is that, given
a small amount of living matter in a suitable environment, the
total amount will quickly increase. A pair of rabbits in Australia
quickly become many tons of rabbit. A few measles bacilli in
a child quickly become many millions. A few seeds dropped by
birds on Krakatoa after volcanic devastation quickly became
luxuriant vegetation. So far as animals are concerned, this property
of living matter is not fully exhibited, since animals require food
that is already organic; but plants can transform inorganic sub-
stances into living matter. This is a purely chemical process, but
it is one from which, presumably, most of the other peculiarities
of living matter, considered as a whole, in some sense follow.

It is an essential feature of living matter that it is not chemically
static, but is undergoing continual chemical change; it is, one
may say, a natural chemical laboratory. Our blood undergoes one
kind of change as it circulates round the body, and an opposite
change when it comes in contact with air in the lungs. Food,
from the moment of contact with the saliva, undergoes a series
of elaborate processes, which end by giving it the chemical
structure appropriate to some part of the body.

There is no reason, except the great complexity of the molecules
that compose a living body, why such molecules should not be
manufactured artificially; nor is there the slightest reason for
supposing that, if they were manufactured, they would lack any-
thing distinctive of living matter naturally generated. Aristotle
thought that there was a vegetable soul in every plant or animal,
and something similar has been widely believed by vitalists. But
for this view there has come to be less and less plausibility as
organic chemistry has progressed. The evidence, though not
conclusive, tends to show that everything distinctive of living
matter can be reduced to chemistry, and therefore ultimately to
physics. The fundamental laws governing living matter are, in

BIOLOGICAL EVOLUTION

all likelihood, the very same that govern the behaviour of the
hydrogen atom, namely the laws of quantum mechanics.

One of the characteristics of living organisms that have seemed
mysterious is the power of reproduction. Rabbits generate rabbits,
robins generate robins, and worms generate worms. Development
from an embryo does not occur in the simplest forms of life;
unicellular organisms merely grow till they reach a certain size,
and then split. Something of this survives in sexual reproduction:
part of the female body becomes an ovum, part of the male body
a sperm, but this part is so much less than half that it seems
qualitatively, and not merely quantitatively, different from the
process of splitting into two equal halves. It is not in the splitting,
however, that the novelty consists, but in the combination of male
and female elements to make a new organism, which, in the
natural process of growth, becomes, in time, like its adult parents.

As a consequence of the Mendelian theory, the process of
heredity has come to be more or less understood. It appears that
in the ovum and in the sperm there are a certain fairly small
number of “genes”, which carry the hereditary characteristics.
The laws of heredity, like those of quantum theory, are discrete
and statistical; in general, when grandparents differ in some
character, we cannot tell which grandparent a given child will
resemble, but we can tell the proportion, out of a large number,
that will resemble this one or that as regards the character in
question.

In general, the genes carry the parental character, but some-
times there are sports, or “mutants”, which differ substantially
from the parent. They occur naturally in a small proportion of
cases, and they can be produced artificially by X-rays. It is these
sports that give the best opportunity for evolution, i.e. for the
development of new kinds of animals or plants by descent from
old kinds.

The general idea of evolution is very old; it is already to be
found in Anaximander (sixth century B.C.), who held that men
are descended from fishes. But Aristotle and the Church banished
such theories until the eighteenth century. Already Descartes,
Kant, and Laplace had advocated a gradual origin for the solar
system, in place of sudden creation followed by a complete absence
of change. As soon as geologists had succeeded in determining
the relative ages of different strata, it became evident from fossils

HUMAN knowledge: its scope and limits

planets. Life, therefore, is almost certainly a very rare phenomenon.
Even on the earth it is transitory : at first the earth was too hot,
and in the end it will be too cold. Some highly conjectural dates
are suggested in Spencer Jones’s Worlds Without End (p. 19). The
age of the earth is probably less than 3,000 million years; the
beginnings of life may be placed at about 1,700 million years ago.
Mammals began about 60 million years ago ; anthropoid apes about
8 million, man about 1 million. It is probable that all forms of
life on earth have evolved from unicellular organisms. How these
were first formed we do not know, but their origin is no more
mysterious than that of helium atoms. There is no reason to
suppose living matter subject to any laws other than those to
which inanimate matter is subject, and considerable reason to
think that everything in the behaviour of living matter is theoreti-
cally explicable in terms of physics and chemistry.

Chapter V

THE PHYSIOLOGY OF SENSATION AND
VOLITION

F rom the standpoint of orthodox psychology, there are two
boundaries between the mental and physical, namely sen-
sation and volition. (“Sensation” may be defined as the
first mental effect of a physical cause^^^voTition^ as the last mental
[caus e of a physical effect .yf am not maintaining that these
definitions will prove ultimately satisfactory, but only that they
may be adopted as a guide in our preliminary survey. In the
present chapter I shall not be concerned with either sensation
or volition themselves, since they belong to psychology; I shall
be concerned only with (t he phy siological antecedents a nd con-
comitants of sensation, and (with the physiological concomitants
and" consequents of volition. Before considering what science’has
tcTsayT irwttttye worth while to look at the matter first from a
common-sense point of view.

Suppose something is said to you, and in consequence you take
some action; for example, you may be a soldier obeying the word
of comman^Physics studies the sound waves that travel through
the air until tney reach the ear ; physiology studies the consequent
event in the ear and nerves and brain, up to the moment when
you hear the sound ; psychology studies the sensation of hearing
and the consequent volition; physiolo gy then resumes the study
of the process, and considers the outgoing chain of events from
the brain to the muscles and the bodily movement expressing
the volition; from that point^pnward, what happens is again part
of the subject-matter of physi cs. The problem of the relation of
mind and matter, which is part of the stock in trade of philosophy,
comes to a head in the transition from events in the brain to the
sensation, jpid from the volition to other events in the brain. It
is thus a two-fo ld probl em: (how does matter affect mind in
sensation, ^and^iow does mind affect matter in volitional do not
propose to consider this problem at this stage ; I mention it now
only to show the relevance of certain parts of physiology to
questions which philosophy must discuss.

The physiological processes which precede and accompany

5 *

human knowledge: its scope and limits

sensation are admirably set forth in Adrian’s book The Basis of
Sensation: The Action of the Sense Organs (London, 1928). As
every one knows, there are two sorts of nerve fibres, those that
carry messages into the brain, and those that carry messages out
of it. The former alone are concerned in the physiology of sen-
sation. Isolated nerves can be stimulated artificially by an electric
current, and there is good reason to believe that the processes
thus set up are essentially similar to those set up naturally in
nerves that are still in place in a living body. When an isolated
nerve is thus stimulated in an adequate manner, a disturbance
is set up which travels along the nerve at a speed of about 220
miles an hour (100 metres a second). Each nerve consists of a
bundle of nerve fibres running from the surface of the body to
the brain or the spinal chord. The nerve fibres which carry mes-
sages to the brain are called “afferen t”, those which carry messages
from the brain are called “ effere nt”. A nerve usually contains both
afferent and efferent fibres, firoadly speaking, the afferent fibres
start from sense-organs and the efferent fibres end in muscles.

The response of a nerve fibre to a stimulus is of what is called
the “all-or-nothing” type, like the response of a gun to pressure
on the trigger. A slight pressure on the trigger produces no result,
but a pressure which is sufficiently great produces a specific result
which is the same however great the pressure may be (within
limits). Similarly when a nerve fibre is stimulated very slightly,
or for a very brief period (less than • 00001 of a second), there
is no result, but when the stimulus is sufficient a current travels
along the nerve fibre for a very brief period (a few thousandths
of a second), after which the nerve fibre is “tired” and will not
transmit another current until it is rested. At first, for two or
three thousandths of a second, the nerve fibre is completely
refractory; then it recovers gradually. During the period of
recovery a given stimulus produces a smaller response, and one
which travels more slowly. Recovery is complete after about a
tenth of a second. The result is that a constant stimulus does not
produce a constant state of excitement in the nerve fibres, but
a series of responses with quiescent periods between. The mes-
sages that reach the brain are, as Adrian puts it, like a stream
of bullets from a machine gun, not like a continuous stream of
water.

It is supposed that in the brain, or the spinal column, there

THE PHYSIOLOGY OF SENSATION AND VOLITION

is a converse mechanism which reconverts the discrete impulses
into a continuous process, but this, so far, is purely hypothetical.

Owing to the discontinuous nature of the response to a stimulus,
the response will be exactly the same to a constant stimulus as to
one which is intermittent with a frequency adapted to the period
of recovery in the nerve. It would seem to follow that there can
be no means of knowing whether the stimulus is constant or
intermittent. But this is not altogether true. Suppose, for instance,
that you are looking at a bright spot of light : if you could keep
your eyes absolutely fixed, your sensations would be the same
if the light flickered with appropriate rapidity as they would be
if the light were steady. But in fact it is impossible to keep the
eyes quite still, and therefore fresh unfatigued nerves are per-
petually being brought into play.

A remarkable fact, which might seem to put a limit on the
informative value of sensations, is, that the response of the nerve
fibre ip the same to any stimulus of sufficient strength and dura-
tion: there is just one message, and only one, that a given nerve
fibre can transmit. But consider the analogy of a typewriter:
if you press a given letter, only one result occurs, and yet the
typewriter as a whole can transmit any information, however
complicated.

The mechanism of the efferent nerve fibres appears to be just
the same as that of the afferent nerve fibres ; the messages that
travel from the brain to the muscles have the same jerky character
as those that travel from the sense-organs to the brain.

But the most interesting question remains: what goes on in
the brain between the arrival of a message by the afferent nerves
and the departure of a message by the efferent nerves ? Suppose
you read a telegram saying “all your property has been destroyed
in an earthquake”, and you exclaim “heavens! I am ruined”.
We feel, rightly or wrongly, that we know the psychological links,
after a fashion, by introspection, but everybody is agreed that
there must also be physiological links. The current brought into
the vision centre by the optic nerve must pass thence to the speech
centre, and then stimulate the muscles which produce your
exclamation. How this happens is still obscure. But it seems clear
that, from a physiological point of view, there is a unitary process
from the physical stimulus to the muscular response. In man this
process may be rendered exceedingly complex by the operation

HUMAN knowledge: its scope and limits

they are taught by different men. What physicists have to teach
is fairly clear, but what have the psychologists to teach ?

There are those among psychologists who take a view which
really denies to psychology the status of a separate science. Accord-
ing to this school, psychology consists in the study of human
and animal behaviour, and the only thing that distinguishes it
from philosophy is its interest in the organism as a whole. The
observations upon which the psychologist must rely, according to
this view, are such as a man might make on animals other than
himself; there is no science, say the adherents of this school,
which has data that a man can only obtain by observation of him-
self. While I admit the importance of what has been learnt by
studying behaviour, I cannot accept this view. There are — and I
am prepared to maintain this dogmatically — many kinds of events
that I can observe when they happen to me, but not when they
happen to any one else. I can observe my own pains and pleasures,
my perceptions, my desires, my dreams. Analogy leads me to
believe that other people have similar experiences, but this is an
inference, not an observation. The dentist does not feel my
toothache, though he may have admirable inductive grounds for
believing that I do.

This suggests a possible definition of psychology, as the science
of those occurrences which, by their very nature, can only be
observed by one person. Such a definition, however, unless
somewhat limited, will turn out to be too wide in one direction,
while too narrow in another. When a number of people observe a
public event, such as the bursting of a rocket or a broadcast by
the Prime Minister, they do not all see or hear exactly the same
thing: there are differences due to perspective, distance from the
source of the sight or sound, defects in the sense-organs, and so
on. Therefore if we were to speak with pedantic accuracy, we
should have to say that everything that can be observed is private
to one person. There is often, however, such a close similarity
between the simultaneous percepts of different people that the
minute differences can, for many purposes, be ignored; we then
say that they are all perceiving the same occurrence, and we place
this occurrence in a public world outside all the observers. Such
occurrences are the data of physics, while those that have not this
social and public character supply (so I suggest) the data of
psychology.

THE SCIENCE OF MIND

According to this view, a datum for physics is something
abstracted from a system of correlated psychological data. When
a crowd of people all observe a rocket bursting, they will ignore
whatever there is reason to think peculiar and personal in their
experience, and will not realize without an effort that there is any
private element in what they see. But they can, if necessary,
become aware of these elements. One part of the crowd sees the
rocket on the right, one on the left, and so on. Thus when each
person's perception is studied in its fullness, and not in the
abstract form which is most convenient for conveying information
about the outside world, the perception becomes a datum for
psychology.

But although every physical datum is derived from a system of
psychological data, the converse is not the case. Sensations
resulting from a stimulus within the body will naturally not be
felt by other people ; if I have a stomach-ache I am in no degree
surprised to find that others are not similarly afflicted. There are
afferent nerves from the muscles, which cause sensations when
the muscles are used; these sensations, naturally, are only felt by
the person concerned. It is only when the stimulus is outside the
body of the percipient, and not always even then, that the sensa-
tion is one of a system which together constitutes one datum for
physics. If a fly is crawling on your hand, the visual sensations
that it causes are public, but the tickling is private. Psychology is
the science which deals with private data, and with the private
aspects of data which common sense regards as public.

To this definition a fundamental objection is raised by a certain
school of psychologists, who maintain that “introspection" is not
a valid scientific method, and that nothing can be scientifically
known except what is derived from public data. This view seems
to me so absurd that if it were not widely held I should ignore
it; but as it has become fashionable in various circles I shall state
my reasons for rejecting it.

To begin with, we need a more precise definition of “public"
and “private" data. “Public" data, for the purpose of those who
reject introspection, are not only data which in fact are shared
by other observers, but also those which might be so shared
given suitable circumstances. Robinson Crusoe, on this view, is
not being unscientifically introspective when he describes the
crops he raised, although there is no other observer to confirm his

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

narrative, for its later parts are confirmed by Man Friday, and
its earlier parts might have been. But when he relates how he
became persuaded that his misfortunes were a punishment for
his previous sinful life, he is either saying something meaningless
or telling what words he would have uttered if he had had any one
to speak to — for what a man says is public, but what he thinks is
private. To maintain that what he says expresses his thought is,
according to this school, to say something not scientifically
verifiable and therefore something which science should not say.
To attempt — as Freud did — to make a science of dreams is a
mistake; we cannot know what a man dreams, but only what he
says he dreams. What he says he dreams is part of physics,
since the saying consists of movements of lips and tongue and
throat; but it is a wanton assumption to suppose that what he
says in professing to relate his dream expresses an actual
experience.

We shall have to define a “public” datum as one which can be
observed by many people, provided they are suitably placed.
They need not all observe it at once, provided there is reason to
think that there has been no change meanwhile: two people
cannot look down a microscope at the same time, but the enemies
of introspection do not mean to exclude data obtained by means
of microscopes. Or consider the fact that, if you press one eyeball
upwards, everything looks double. What is meant by saying that
things “look” double? This can only be interpreted by distinguish-
ing between the visual perception and the physical fact, or else by
a subterfuge. You may say: “When I say that Mr. A. is seeing
double, I say nothing about his perceptions; what I say means:
Tf Mr. A. is asked, he will say he is seeing double*.” Such an
interpretation makes it meaningless to inquire whether Mr. A. is
speaking the truth, and impossible to discover what it is that he
thinks he is asserting.

Dreams are perhaps the most indubitable example of facts
which can only be known by means of private data. When I
remember a dream I can relate it, either truly or with embellish-
ments; I can know which I am doing, but others seldom can. I
knew a Chinese lady who, after a few lessons in psycho-analysis,
began to have perfect text-book dreams; the analyst was de-
lighted, but her friends were sceptical. Although no one except
the lady could be sure of the truth, I maintain that the fact as to

THE SCIENCE OF MIND

what she had dreamed was just as definitely such-and-such rather
than so-and-so as in the case of a physical phenomenon.

We shall have to say: A “public” datum is one which generates
similar sensations in all percipients throughout a certain space-
time region, which must be considerably larger than the region
occupied by one human body throughout (say) half a second —
or rather, it is one which would generate such sensations if
suitably placed percipients were present (this is to allow for
Robinson Crusoe’s crops).

This distinction between public and private data is one which
it is difficult to make precise. Roughly speaking, sight and hearing
give public data, but not always. When a patient is suffering from
jaundice everything looks yellow, but this yellowness is private.
Many people are liable to a buzzing in the ears which is sub-
jectively indistinguishable from the hum of telegraph wires in a
wind. The privacy of such sensations is only known to the
percipient through the negative testimony of other people. Touch
gives public data in a sense, since different people can successively
touch the same object. Smells can be so public as to become
grounds of complaint to the sanitary authority. Tastes are public
in a lesser degree, for, though two people cannot eat the same
mouthful, they can eat contiguous portions of the same viand ; but
the curate’s egg shows that this method is not quite reliable. It
is, however, sufficiently reliable to establish a public distinction
between good cooks and bad ones, though here introspection
plays an essential part, for a good cook is one who causes
pleasure to most consumers, and the pleasure of each is purely
private.

I have kept this discussion on a common-sense level, but at a
later stage I shall resume it, and try to probe more deeply into
the whole question of private data as a basis for science. For the
present I am content to say that the distinction between public
and private data is one of degree, that it depends upon testimony
which bears witness to the results of introspection, that physiology
would lead to the expectation that sensations caused by a stimulus
inside a human body would be private, and, finally, that many of
the facts of which each one of us is most certain are known to us
by means private to ourselves. Do you like the smell of rotten
eggs ? Are you glad the war is over ? Have you a toothache ? These
questions are not difficult for you to answer, but no one else can

HUMAN knowledge: its scope and limits

answer them except by inferences from your behaviour, including
your testimony.

I conclude, therefore, that there is knowledge of private data,
and that there is no reason why there should not be a science of
them. This being granted, we can now inquire what psychology
in fact has to say.

There is, to begin with, a matter of which the importance is
often overlooked, and that is the correlation of physical occurrences
with sensation. Physicists and astronomers base their assertions
as to what goes on in the outer world upon the evidence of the
senses, especially the sense of sight. But not a single one of the
occurrences that we are told take place in the physical world is a
sensation; how, then, can sensations confirm or confute a physical
theory? Let us take an illustration belonging to the infancy of
science. It was early discovered that an eclipse of the sun is due
to the interposition of the moon, and it was found that eclipses
could be predicted. Now what was directly verified when an
eclipse occurred was a certain sequence of expected sensations.
But the development of physics and physiology has gradually
caused a vast gulf between the sensations of an astronomer watching
an eclipse and the astronomical fact which he infers. Photons start
from the sun, and when the moon is not in the way some of them
reach an eye, where they set up the kind of complicated process
that we considered in the last chapter; at last, when the process
reaches the astronomer’s brain, the astronomer has a sensation.

The sensation can only be evidence of the astronomical fact if
laws are known connecting the two, and the last stage in these
laws must be one connecting stimulus and sensation, or connecting
occurrences in the optic nerve or the brain with sensation. The
sensation, it should be observed, is not at all like the astronomical
fact, nor are the two necessarily connected. It would be possible
to supply an artificial stimulus causing the astronomer to have
an experience subjectively indistinguishable from what we call
“seeing the sun”. And at best the resemblance between the
sensation and the astronomical fact cannot be closer than that
between a gramophone record and the music that it plays, or
between a library catalogue and the books that it enumerates. It
follows that, if physics is an empirical science, whose statements
can be confirmed or confuted by observation, then physics must
be supplemented by laws connecting stimulus and sensation.

6 z

THE SCIENCE OF MIND

Now such laws belong to psychology. Therefore what is em-
pirically verifiable is not pure physics in isolation, but physics
plus a department of psychology. Psychology, accordingly, is an
essential ingredient in every part of empirical science.

So far, however, we have not inquired whether there are any
laws that connect one mental event with another. The laws of
correlation so far considered have been such as connect a physical
stimulus with a mental response; what we have now to consider
is whether there are any causal laws which are entirely within one
mind. If there are, psychology is to that extent an autonomous
science. The association of ideas, as it appears for example in
Hartley and Bentham, was a law of this kind, but the conditioned
reflex and the law of habit, which have taken its place, are
primarily physiological and only derivatively psychological, since
association is thought to be caused by the creation of paths in the
brain connecting one centre with another. We may still state the
association of ideas in purely psychological terms, but when so
stated it is not a law as to what always happens, but only as to
what is apt to happen. It has not therefore the character that
science hopes to find in a causal law, or at least used to hope for
before the rise of quantum theory.

The same thing may be said of psycho-analysis, which aims
at discovering purely mental causal laws. I do not know of any
psycho-analytic law which professes to say what will always
happen in such and such circumstances. When a man, for
example, suffers from claustrophobia, psycho-analysis will
discover this or that past experience which is held to explain
his trouble; but many people will have had the same experience
without the same result. The experience in question, accordingly,
though it may well be part of the cause of the phobia, cannot
be its whole cause. We cannot, this being the case, find in psycho-
analysis any examples of purely psychical causal laws.

In the last chapter we suggested, as a probable hypothesis,
the view that all bodily behaviour is theoretically explicable in
physical terms, without taking any account of the mental
concomitants of physiological occurrences. This hypothesis, it
should be observed, in no way decides our present question.
If A and B are two events in the brain, and if A causes B, then
if a is a mental concomitant of A, and b of B, it will follow that
a causes A, which is a purely mental causal law. In fact, causal

human knowledge: its scope and limits

laws are not of the simple form “A causes B”, but in their true
form the principle remains the same.

Although, at present, it is difficult to give important examples
of really precise mental causal laws, it seems pretty certain, on
a common-sense basis, that there are such laws. If you tell a
man that he is both a knave and a fool, he will be angry; if you
inform your employer that he is universally regarded as a
swindler and a bloodsucker, he will invite you to seek employment
elsewhere. Advertising and political propaganda supply a mass
of materials for the psychology of belief. The feeling one has
in a novel or a play as to whether the behaviour of the characters
is “right” is based upon unformulated knowledge of mental
causality, and so is shrewdness in handling people. In such cases,
the knowledge involved is pre-scientific, but it could not exist
unless there were scientific laws which could be ascertained by
sufficient study.

There are a certain number of genuine causal laws of the
kind in question, though so far they are mostly concerned with
matters that have no great intrinsic interest. Take, for example,
after-images: you look fixedly at a bright red object, and then
shut your eyes; you see first a gradually fading red image, and
then a green image, of approximately the same shape. This is
a law for which the evidence is purely introspective. Or again,
take a well-known illusion :

In the figure the two horizontal lines are parallel, but they look
as if they approached each other towards the right. This again
is a law for which the evidence is purely introspective. In both
cases there are physiological explanations, but they do not
invalidate the purely psychological laws.

THE SCIENCE OF MIND

I conclude that, while some psychological laws involve
physiology, others do not. Psychology is a science distinct from
physics and physiology, and in part independent of them. All
the data of physics are also data of psychology, but not vice
versa ; data belonging to both are made the basis of quite different
inferences in the two sciences. Introspection is valid as a source
of data, and is to a considerable extent amenable to scientific
controls.

There is much in psychology that is genuinely scientific
although it lacks quantitative precision. Take, for example, the
analysis of our spatial perceptions, and the building up of the
common-sense notion of space from its sensational foundations.
Berkeley’s theory of vision, according to which everything looks
flat, is disproved by the stereoscope. The process by which we
learn in infancy to touch a place that we see can be studied by
observation. So can volitional control: a baby a few months old
can be watched learning with delight to move its toes at will,
instead of having to look on passively while they wriggle in
purely reflex movements. When, in later life, you acquire some
skill, such as riding a bicycle, you find yourself passing through
stages: at first you will certain movements of your own body,
in the hope that they will cause the desired movements of the
bicycle, but afterwards you will the movements of the bicycle
directly, and the necessary movements of your body result auto-
matically. Such experiences throw much light on the psychology
of volition.

Much psychology is involved in connecting sensory stimuli
with the beliefs to which they give rise. I am thinking of such
elementary occurrences as thinking “there’s a cat” when certain
coloured patches in motion pass across your field of vision. It
is obvious that the same sensory stimulus could be caused other-
wise than by a cat, and your belief would then be false. You
may see a room reflected in a mirror, and think that it is “real”.
By studying such occurrences we become aware that a very
large part of what we think we perceive consists of habits caused
by past experience. Our life is full of expectations of which,
as a rule, we only become aware when they are disappointed.
Suppose you see half of a horse that is just coming round a
corner; you may be very little interested, but if the other half
proved to be cow and not horse you would experience a shock

65 E

HUMAN knowledge: its scope and limits

of surprise which would be almost unendurable. Yet it must be
admitted that such an occurrence is logically possible.

The connection of pleasure and pain and desire with habit-
formation can be studied experimentally. Pavlov, whose work
nowhere appeals to introspection, put a dog in front of two doors,
on one of which he had drawn an ellipse and on the other a
circle. If the dog chose the right door he got his dinner; if he
chose the wrong one he got an electric shock. Thus stimulated,
the dog’s progress in geometry was amazingly rapid. Pavlov
gradually made the ellipse less and less eccentric, but the dog
still distinguished correctly, until the ratio of minor to major
axis was reduced to 8:9, when the poor beast had a nervous
breakdown. The utility of this experiment in connection with
schoolboys and criminals is obvious.

Or, again, take the question: why do we believe what we do?
In former times, philosophers would have said it was because
God had implanted in us a natural light by which we knew the
truth. In the early nineteenth century they might have said it
was because we had weighed the evidence and found a preponder-
ance on one side. But if you ask a modern advertiser or political
propagandist he will give you a more scientific and more
depressing answer. A large proportion of our beliefs are based
on habit, conceit, self-interest, or frequent iteration. The advertiser
relies mainly on the last of these, but if he is clever he combines
it skilfully with the other three. It is hoped that by studying
the psychology of belief, those who control propaganda will in
time be able to make anybody believe anything. Then the
totalitarian State will become invincible.

In regard to human knowledge there are two questions that
may be asked: first, what do we know? and second, how do we
know it? The first of these questions is answered by science,
which tries to be as impersonal and as dehumanized as possible.
In the resulting survey of the universe it is natural to start with
astronomy and physics, which deal with what is large and what
is universal; life and mind, which are rare and have, apparently,
little influence on the course of events, must occupy a minor
position m this impartial survey. But in relation to our second
question, namely, how do we come by our knowledge, psychology
is the most important of the sciences. Not only is it necessary
to study psychologically the processes by which we draw inferences,

THE SCIENCE OF MIND

but it turns out that all the data upon which our inferences
should be based are psychological in character, that is to say,
they are experiences of single individuals. The apparent publicity
of our world is in part delusive and in part inferential; all the
raw material of our knowledge consists of mental events in the
lives of separate people. In this region, therefore, psychology is
supreme.

PART II

LANGUAGE

Chapter I

THE USES OF LANGUAGE

L anguage, like other things of mysterious importance, such
as breath, blood, sex, and lightning, has been viewed
superstitiously ever since men were capable of recording
their thoughts. Savages fear to disclose their true name to an
enemy, lest he should work evil magic by means of it. Origen
assures us that pagan sorcerers could achieve more by using
the sacred name Jehovah than by means of the names Zeus,
Osiris, or Brahma. Familiarity makes us blind to the linguistic
emphasis in the Commandment: “Thou shalt not take the name
of the Lord in vain.” The habit of viewing language superstitiously
is not yet extinct. “In the beginning was the Word”, says our
version of St. John’s Gospel, and in reading some logical
positivists I am tempted to think that their view is represented
by this mistranslated text.

Philosophers, being bookish and theoretical folk, have been
interested in language chiefly as a means of making statements
and conveying information, but this is only one of its purposes,
and perhaps not the most primitive. What is the purpose of
language to a sergeant-major? On the one hand there is the
language of words of command, designed to cause identical
simultaneous bodily movements in a number of hearers ; on the
other hand there is bad language, designed to cause humility
in those in whom the expected bodily movements have not been
caused. In neither case are words used, except incidentally, to
state facts or convey information.

Language can be used to express emotions, or to influence
the behaviour of others. Each of these functions can be performed,
though with less adequacy, by pre-linguistic methods. Animals
emit shrieks of pain, and infants, before they can speak, can express
rage, discomfort, desire, delight, and a whole gamut of feelings,
by cries and gurgles of different kinds. A sheep dog emits imper-
atives to his flock bv means hardly distinguishable from those
that the shepherd employs towards him. Between such noises
and speech no sharp line can be drawn. When the dentist hurts
you, you may emit an involuntary groan; this does not count as

human knowledge: its scope AND LIMITS

speech. But if he says “let me know if I hurt you”, and you
then make the very same sound, it has become speech, and
moreover speech of the sort intended to convey information.
This example illustrates the fact that, in the matter of language
as in other respects, there is a continuous gradation from animal
behaviour to that of the most precise man of science, and from
pre-linguistic noises to the polished diction of the lexicographer.

A sound expressive of emotion I shall call an “interjetion”.
Imperatives and interjections can already be distinguished in
the noises emitted by animals. When a hen clucks at her brood
of chickens, she is uttering imperatives, but when she squawks
in terror she is expressing emotion. But as appears from your
groan at the dentist s, an interjection may convey information,
and the outside observer cannot tell whether or not it is intended
to do so. Gregarious animals emit distinctive noises when they
find food, and other members of the herd are attracted when
they hear these noises, but we cannot know whether the noises
merely express pleasure or are also intended to state “food here”.

Whenever an animal is so constructed that a certain kind of
circumstance causes a certain kind of emotion, and a certain
kind of emotion causes a certain kind of noise, the noise conveys
to a suitable observer two pieces of information, first, that the
animal has a certain kind of feeling, and second, that a certain
kind of circumstance is present. The sound that the animal
emits is public, and the circumstance may be public — e.g. the
presence of a shoal of fish if the animal is a sea-gull. The animal’s
cry may act directly on the other members of its species, and we
shall then say that they “understand” its cry. But this is to suppose
a “mental” intermediary between the hearing of the cry and the
bodily reaction to the sound, and there is no real reason to suppose
any such intermediary except when the response is delayed.
Much of the importance of language is connected with delayed
responses, but I will not yet deal with this topic.

Language has two primary purposes, expression and com-
munication. In its most primitive forms it differs little from some
other forms of behaviour. A man may express sorrow by sighing,
or by saying “alas!” or “woe is me!” He may communicate by
pointing or by saying “look”. Expression and communication
are not necessarily separated; if you say “look” because you see
a ghost, you may say it in a tone that expresses horror. This

7 *

THE USES OF LANGUAGE

applies not only to elementary forms of language; in poetry, and
especially in songs, emotion and information are conveyed by
the same means. Music may be considered as a form of language
in which emotion is divorced from information, while the telephone
book gives information without emotion. But in ordinary speech
both elements are usually present.

Communication does not consist only of giving information;
commands and questions must be included. Sometimes the two
are scarcely separable: if you are walking with a child, and you
say “there’s a puddle there”, the command “don’t step in it”
is implicit. Giving information may be due solely to the fact that
the information interests you, or may be designed to influence
behaviour. If you have just seen a street accident, you will wish
to tell your friends about it because your mind is full of it ; but
if you tell a child that six times seven is forty-two you do so
merely in the hope of influencing his (verbal) behaviour.

Language has two interconnected merits: first, that it is social,
and second that it supplies public expression for “thoughts”
which would otherwise remain private. Without language, or
some pre-linguistic analogue, our knowledge of the environment
is confined to what our own senses have shown us, together with
such inferences as our congenital constitution may prompt;
but by the help of speech we are able to know what others can
relate, and to relate what is no longer sensibly present but only
remembered. When we see or hear something which a companion
is not seeing or hearing, we can often make him aware of it by the
one word “look” or “listen”, or even by gestures. But if half an
hour ago we saw a fox, it is not possible to make another person
aware of this fact without language. This depends upon the
fact that the word “fox” applies equally to a fox seen or a fox
remembered, so that our memories, which in themselves are
private, are represented to others by uttered sounds, which are
public. Without language, only that part of our life which consists
of public sensations would be communicable, and that only to
those so situated as to be able to share the sensations in question.

It will be seen that the utility of language depends upon the
distinction between public and private experiences, which is
important in considering the empirical basis of physics. This
distinction, in turn, depends partly on physiology, partly on the
persistence of sound-waves and light quanta, which makes

HUMAN knowledge: its scope and limits

possible the two forms of language, speech and writing. Thus
language depends upon physics, and could not exist without the
approximately separable causal chains which, as we shall see,
make physical knowledge possible, and since the publicity of
sensible objects is only approximate, language applying to them,
considered socially, must have a certain lack of precision. I need
hardly say that I am not asserting that the existence of language
requires a knowledge of physics. What I am saying is that language
would be impossible if the physical world did not in fact have
certain characteristics, and that the theory of language is at certain
points dependent upon a knowledge of the physical world.
Language is a means of externalizing and publicizing our own
experiences. A dog cannot relate his autobiography; however
eloquently he may bark, he cannot tell you that his parents were
honest though poor. A man can do this, and he does it by
correlating “thoughts” with public sensations.

Language serves not only to express thoughts, but to make
possible thoughts which could not exist without it. It is sometimes
maintained that there can be no thought without language,
but to this view I cannot assent : I hold that there can be thought,
and even true and false belief, without language. But however
that may be, it cannot be denied that all fairly elaborate thoughts
require words. I can know, in a sense, that I have five fingers,
without knowing the word “five”, but I cannot know that the
population of London is about eight millions unless I have
acquired the language of arithmetic, nor can I have any thought
at all closely corresponding to what is asserted in the sentence : “the
ratio of the circumference of a circle to the diameter is approxi-
mately 3-14159”. Language, once evolved, acquires a kind of
autonomy : we can know, especially in mathematics, that a sentence
asserts something true, although what it asserts is too complex
to be apprehended even by the best minds. Let us consider for
a moment what happens psychologically in such cases.

In mathematics, we start from rather simple sentences which
we believe ourselves capable of understanding, and proceed, by
rules of inference which we also believe ourselves to understand,
to build up more and more complicated symbolic statements,
which, if our initial assumptions are true, must be true whatever
they may mean. As a rule it is unnecessary to know what they
“mean”, if their “meaning” is taken to be a thought which

THE USES OF LANGUAGE

might occur in the mind of a superhuman mathematical genius.
But there is another kind of “meaning”, which gives occasion
for pragmatism and instrumentalism. According to those who
adopt this view of “meaning”, what a complicated mathematical
sentence does is to give a rule for practical procedure in certain
kinds of cases. Take, for instance, the above statement about the
ratio of the circumference of a circle to the diameter. Suppose
you are a brewer, and you desire hoops of a given diameter for
your beer barrels, then the sentence gives you a rule by which
you can find out how much material you will need. This rule
may consist of a fresh sentence for each decimal point, and there
is therefore no need ever to grasp its significance as a whole.
The autonomy of language enables you to forego this tedious
process of interpretation except at crucial moments.

There are two other uses of language that are of great import-
ance; it enables us to conduct our transactions with the outer
world by means of symbols that have (l) a certain degree of
permanence in time, (2) a considerable degree of discreteness in
space. Each of these merits is more marked in writing than in
speech, but is by no means wholly absent in speech. Suppose
you have a friend called Mr. Jones. As a physical object his
boundaries are somewhat vague, both because he is continually
losing and acquiring electrons, and because an electron, being a
distribution of energy, does not cease abruptly at a certain distance
from its centre. The surface of Mr. Jones, therefore, has a certain
ghostly impalpable quality, which you do not like to associate
with your solid-seeming friend. It is not necessary to go into the
niceties of theoretical physics in order to show that Mr. Jones is
sadly indeterminate. When he is cutting his toe nails, there is
a finite time, though a short one, during which it is doubtful
whether the parings are still part of him or not. When he eats
a mutton chop, at what moment does it become part of him?
When he breathes out carbon dioxide, is the carbon part of him
until it passes his nostrils ? Even if we answer in the affirmative,
there is a finite time during which it is questionable whether
certain molecules have or have not passed beyond his nostrils.
In these and other ways, it is doubtful what is part of Mr. Jones
and what is not. So much for spatial vagueness.

There is the same problem as regards time. To the question
“what are you looking at ?” you may answer “Mr. Jones”, although

HUMAN knowledge: its scope and limits

at one time you see him full-face, at another in profile, and at
another from behind, and although at one time he may be running
a race and at another time dozing in an arm-chair. There is
another question, namely “what are you thinking of?” to which
you may also answer “Mr. Jones”, though what is actually in
your mind may be very different on different occasions : it may be
Mr. Jones as a baby, or Mr. Jones being cross because his break-
fast is late, or Mr. Jones receiving the news that he is to be
knighted. What you are experiencing is very different on these
various occasions, but for many practical purposes it is convenient
to regard them as all having a common object, which we suppose
to be the meaning of the name “Mr. Jones”. This name, especially
when printed, though it cannot wholly escape the indefiniteness
and transience of all physical objects, has much less of both than
Mr. Jones has. Two instances of the printed words “Mr. Jones”
are much more alike than (for instance) the spectacle of Mr.
Jones running and the memory of Mr. Jones as a baby. And each
instance, if printed, changes much more slowly than Mr. Jones
does: it does not eat or breathe or cut its toe nails. The name,
accordingly, makes it much easier than it would otherwise be to
think of Mr. Jones as a single quasi-permanent entity, which,
though untrue, is convenient in daily life.

Language, as appears from the above discussion of Mr. Jones,
though a useful and even indispensable tool, is a dangerous one,
since it begins by suggesting a definiteness, discreteness, and
quasi-permanence in objects which physics seems to show that
they do not possess. The philosopher, therefore, is faced with the
difficult task of using language to undo the false beliefs that it
suggests. Some philosophers, who shrink from the problems and
uncertainties and complications involved in such a task, prefer
to treat language as autonomous, and try to forget that it is
intended to have a relation to fact and to facilitate dealings with
the environment. Up to a point, such a treatment has great
advantages: logic and mathematics would not have prospered as
they have done if logicians and mathematicians had continually
remembered that symbols should mean something. “Art for
art's sake” is a maxim which has a legitimate sphere in logic
as in painting (though in neither case does it give the whole
truth). It may be that singing began as an incident in courtship,
and that its biological purpose was to promote sexual intercourse ;

THE USES OF LANGUAGE

but this fact (if it be a fact) will not help a composer to produce
good music. Language is useful when you wish to order a meal
in a restaurant, but this fact, similarly, is of no importance to the
pure mathematician.

The philosopher, however, must pursue truth even at the
expense of beauty, and in studying language he must not let
himself be seduced by the siren songs of mathematics. Language,
in its beginnings, is pedestrian and practical, using rough and
ready approximations which have at first no beauty and only a
very limited degree of truth. Subsequent refinements have too
often had aesthetic rather than scientific motives, but from the
inquiry upon which we are about to embark aesthetic motives
must, however reluctantly, be relentlessly banished.

Chapter II

OSTENSIVE DEFINITION

41 stensive definition” may be defined as “any process

I 1 by which a person is taught to understand a word other-
wise than by the use of other words”. Suppose that,
knowing no French, you are shipwrecked on the coast of
Normandy : you make your way into a farmhouse, you see bread
on the table, and, being famished, you point at it with an
inquiring gesture. If the farmer thereupon says “pain 9 \ you will
conclude, at least provisionally, that this is the French for
“bread”, and you will be confirmed in this view if the word is
not repeated when you point at other kinds of eatables. You
will then have learnt the meaning of the word by ostensive
definition. It is clear that, if you know no French and your teacher
knows no English, you must depend upon this process during
your first lessons, since you have no linguistic means of com-
munication.

The process of ostensive definition, however, is better exem-
plified when the learner knows no language at all than when
he already possesses a language of his own. An adult knows that
there are words, and will naturally suppose that the French
have a way of naming bread. His knowledge takes the form:
“ 4 Pain 9 means ‘bread’ ”. It is true that, when you were ship-
wrecked, it was by means of actual bread that you acquired this
knowledge, but if you had been shipwrecked with a dictionary
the actual bread would not have been necessary. There are two
stages in the acquisition of a foreign language, the first that in
which you only understand by translating, the second that in
which you can “think” in the foreign language. In the first stage
you know that “pain” means “bread”, in the second stage you
know that it means bread. The infant, possessing as yet no
language, has to begin with the second stage. His success does
credit to the capacities of the infant mind.

Knowing a language has two aspects, passive and active:
passive when you understand what you hear, active when you
can speak yourself. Dogs to some degree achieve the former,
and children usually achieve it some time before the latter.

OSTENSIVE DEFINITION

Knowing a language does not mean a capacity for explicit explana-
tion of what its words signify; it means that hearing the words
has appropriate effects, and using them has appropriate causes.
I have sometimes, in the course of travel, watched a quarrel
springing up between two men whose language I did not under-
stand, and it was difficult not to feel their mounting excitement
ridiculous. But probably the first was accusing the second of
being the offspring of parents who were not married, and the
second was retorting that the first’s wife was unfaithful. If I had
understood, the effect of the insult and the cause of the retort
would have been obvious. As this example illustrates, a person
knows a language when hearing certain sounds has certain effects
and uttering them has certain causes. The process by which,
in the infant, the establishment of these causal laws is begun,
is the process of ostensive definition.

Ostensive definition, in its earliest form, requires certain
conditions. There must be a feature of the environment which is
noticeable, distinctive, emotionally interesting, and (as a rule)
frequently recurring, and the adult must frequently utter the
name of this feature at a moment when the infant is attending
to it. Of course there are risks of error. Suppose the child has
milk in a bottle. You may each time say “milk” or each time say
“bottle”. In the former case the child may think “milk” is the
right word for a bottle of water; in the latter case, he may think
“bottle” the right word for a glass of milk. To avoid such errors,
you should in theory apply Mill’s inductive canons, remembering
that induction is a bodily habit, and only by courtesy a logical
process. Instead of saying merely “milk” or merely “bottle”,
you should say “bottle of milk”; you should then, on appropriate
occasions, say “glass of milk” and “bottle of water”. In time,
by the use of Mill’s canons, the infant, if he survives, will learn
to speak correctly. But I am not giving practical pedagogic
advice; I am merely exemplifying a theory.

The passive part in ostensive definition is merely the familiar
business of association or the conditioned reflex. If a certain
stimulus A produces in a child a certain reaction R, and is fre-
quently experienced in conjunction with the word B, it will happen
in time that B will produce the reaction R, or some part of it.
As soon as this has happened, the word B has acquired a “mean-
ing” for the child: it “means” A. The meaning may not be quite

human knowledge: its scope and limits

what the adult intended: the adult may have intended “bottle”
and the child may understand the word as meaning milk. But
that does not prevent the child from possessing a word that has
meaning; it only signifies that the child's language is not yet
correct English.

When an experience causes violent emotion, repetition may be
unnecessary. If a child, after learning to understand “milk”,
is given milk so hot as to scald his mouth, and you say “hot”,
he may ever after understand this word. But when an experience
is uninteresting, many repetitions may be necessary.

The active part in the learning of language requires other
capacities, which however, are of less philosophic interest. Dogs
cannot learn human speech because they are anatomically in-
capable of producing the right sounds. Parrots, though they can
produce more or less the right sounds, seem incapable of acquiring
the right associations, so that their words do not have meaning.
Infants, in common with the young of the higher animals, have
an impulse to imitate adults of their own species, and therefore
try to make the sounds that they hear. They may, on occasion,
repeat sounds like a parrot, and only subsequently discover the
“meaning” of the sounds. In that case the sounds cannot count
as words until they have acquired meaning for the child. For
every child it is a discovery that there are words, i.e. sounds
with meaning. Learning to utter words is a joy to the child, largely
because it enables him to communicate his wishes more definitely
than he had been able to do by crying and making gestures. It
is owing to this pleasure that children go through the mental
labour and muscular practice involved in learning to talk.

In general, though not universally, repetition is necessary for
an ostensive definition, for ostensive definition consists in the
creation of a habit, and habits, as a rule, are learned gradually.
The exceptional cases are illustrated by the proverbs “once
bit, twice shy” and “the burnt dblild dreads the fire”. Apart from
such unusually emotional matters, the words that have ostensive
definitions denote frequently recurring features of the environ-
ment, such as the members of the family, foods, toys, pet animals,
etc. This involves the process of recognition, or something of
the kind. Although a child’s mother looks somewhat different
on different occasions, he thinks of her (when he begins to think)
as always the same person, and feels no difficulty in applying the

OSTENSIVE DEFINITION

same name to her various epiphanies. Language, from the start,
or rather from the start of reflection on language, embodies the
belief in more or less permanent persons and things. This is
perhaps the chief reason for the difficulty of any philosophy
which dispenses with the notion of substance. If you were to
tell a child that his mother is a series of sensible impressions,
connected by similarity and causal relations, but without
material identity, and if by a miracle you could make him
understand what you meant, he would consider you demented
and be filled with indignation. The process called “recognition”
is therefore one that demands investigation.

Recognition, as a physiological or psychological occurrence,
may or may not be veridical. It fails in an every-day sense to be
veridical when we mistake one of two twins for the other, but
it may be metaphysically misleading even when it is correct
from the standpoint of common sense. Whether there is anything
identical, and if so what, between two different appearances of
Mr. A, is a dark and difficult question, which I shall consider
in connection with proper names. For the moment I wish to
consider recognition as a process which actually occurs, without
regard to its interpretation.

The first stage in the development of this process is repetition
of a learnt reaction when the stimulus is repeated. It must be a
learnt reaction, since recognition must grow out of a process
involving something, in later reactions to a given stimulus, which
was not present in the first reaction. Suppose, for instance, you
give a child a glass of milk containing bitter medicine: the first
time he drinks the doctored milk and makes a face, but the
second time he refuses the milk. This is subjectively something
like recognition, even if the second time he is mistaken in
supposing that the milk contains medicine. It is clear that this
process may be purely physiological, and that it involves only
similarity, not identity, in stimulus and response. The learning
of words by ostensive definition can be brought wholly within
this primitive stage. The child’s world contains a number of
similar stimuli to which he has learnt to respond by similar
noises, namely those that are instances of the word “milk”;
it contains also another set of similar stimuli to which he has
learnt to respond by instances of the word “mother”. In this
there is nothing involving any beliefs or emotions in the child.

HUMAN knowledge: its scope and limits

It is only as a result of subsequent reflection that the child, now
become a philosopher, concludes that there is one word, “mother”,
and one person, Mother. I believe this first step in philosophy
to be mistaken. The word “mother”, I should say, is not a single
entity, but a class of similar noises; and Mother herself is also
not a single entity, but a class of causally connected occurrences.
These speculations, however, are irrelevant to the process of
ostensive definition, which, as we have just seen, requires only
the very first stage on the road towards what would usually
count as recognition, namely similar learnt responses to similar
stimuli.

This primitive form of recognition is relevant in the analysis
of memory and in explaining the similarity of an idea to an
impression (to borrow Hume’s phraseology). When I remember
a past event, I cannot make it itself occur again, though I may
be able to make a similar event occur. But how do I know that
the new event is similar to the old one ? Subjectively, I can only
know by comparing an idea with an impression: I have an idea
of the past event and an impression of the present event, and
I perceive that they are similar. But this is not sufficient, since
it does not prove that my idea of the past event is similar to my
impression of th t past event when it existed. This, in fact, cannot
be proved, and is in some sense one of the premisses of knowledge.
But although it cannot be strictly proved, it can be in various
ways confirmed. You may describe Mr. A while he is present,
and your description may be recorded on a dictaphone. You
may later describe him from memory, and compare your new
description with the dictaphone record. If they agree closely,
your memory may be accepted as correct.

This illustration depends upon a fact which is fundamental
in this subject, namely, that we apply the same words to ideas
as to the impressions which are their prototypes. This explains
the possibility of learning a word ostensively by means of a single
sensible occurrence. I saw Disraeli once, and once only, and was
told, at the moment, “that’s Dizzy”. I have since very frequently
remembered the occurrence, with the name “Dizzy” as an
essential part of the memory. This has made it possible for a
habit to be formed by repetition of the idea (in Hume’s sense),
although the impression has never been repeated. It is obvious
that ideas differ from impressions in various ways, but their

OSTENSIVE DEFINITION

similarity to their prototypes is vouched for by the fact that
they cause the same words. The two questions, “what are you
looking at?” and “what are you thinking of?” may, on two
different occasions, be answered identically.

Let us consider the different kinds of words that are commonly
learnt by means of ostensive definitions. What I have in mind is
a logical form of the grammatical doctrine of parts of speech.

We have already had occasion for a preliminary consideration
of proper names. I shall say no more about them at present, as
they will be the subject of a separate chapter.

Next come names of species: man, woman, cat, dog, etc. A
species of this sort consists of a number of separate individuals,
having some recognizable degree of likeness to each other. In
biology before Darwin, “species” was a prominent concept.
God had created a pair of each species, and different species
could not interbreed, or, in the exceptional cases when they could,
such as horse and ass, the offspring was sterile. There was an
elaborate hierarchy of genera, families, orders, etc. This kind
of classification, which was and is convenient in biology, was
extended by the scholastics to other regions, and impeded logic
by creating the notion that some ways of classifying are more
correct than others. As regards ostensive definition, different
experiences will produce different results. Most children learn
the word “dog” ostensively; some learn in this way the kinds of
dogs, collies, St. Bernards, spaniels, poodles, etc., while others,
who have little to do with dogs, may first meet with these words
in books. No child learns the word “quadruped” ostensively,
still less the word “animal” in the sense in which it includes
oysters and limpets. He probably learns “ant”, “bee”, and
“beetle” ostensively, and perhaps “insect”, but if so he will
mistakenly include spiders until corrected.

(^Names of substances not obviously collections of individuals,
such as “milk”, “bread”, “wood”, are apt to be learnt ostensively
when they denote things familiar in every-day life. The atomic
theory is an attempt to identify this class of objects with the
former, so that milk, for instance, is a collection of milky
individuals (molecules), just as the human race is a collection of
men, women, and children. J But to unscientific apprehension
such names of substances are not to be assimilated to species
composed of separate individuals.

HUMAN knowledge: its scope and limits

Next come qualities: red, blue, hard, soft, hot, cold, etc.
Many of these are usually learnt ostensively, but the less common
ones, such as vermilion, may be described by their similarities
and differences.

Names of certain relations, such as “up”, “down”, “right”,
“left”, “before”, “after”, are usually learnt ostensively. So are
such words as “quick” and “slow”.

There are a number of words of the sort that I call “ego-
centric”, which differ in meaning according to the speaker and
his position in time and space. Among these the simple ones are
learnt ostensively, for instance “I”, “you”, “here”, “now”.
These words raise problems which we will consider in a later
chapter.

All the words I have mentioned hitherto belong to the public
world. A spectator can see when a certain feature of the public
environment is attracting a child’s attention, and then mention
the name of this feature. But how about private experiences,
such as stomach-ache, pain, or memory? Certainly some words
denoting private kinds of experience are learnt ostensively. This
is because the child shows in behaviour what he is feeling : there
is a correlation between e.g. pain and tears.

There are no definite limits to what can be learnt by ostensive
definition. “Cross”, “crescent”, “swastika” can be learnt in this
way, but not “chiliagon”. But the point where this method of
learning becomes impossible depends upon the child’s experience
and capacity.

The words so far mentioned are all capable of being used as
complete sentences, and are in fact so used in their most primitive
employment. “Mother”, “dog”, “cat”, “milk”, and so on, may
be used alone to express either recognition or desire. “Hard”,
“soft”, “hot”, “cold” would be more naturally used to express
recognition than desire, and usually to express recognition
accompanied by surprise. If the toast is uneatable because it is
old you may say “hard”; if a ginger biscuit has lost its crispness
by exposure to air you may say “soft”. If the bath scalds you,
you say “hot”; if it freezes you, you say “cold”. “Quick” is
frequently used by parents as an imperative; “slow” is used
similarly on roads and railways where there is a curve. The words
“up” and “down” are habitually used as complete sentences
by lift-boys; “in” and “out” are similarly used at turnstiles.

OSTENSIVE DEFINITION

“Before” and “after” are used as complete sentences in advertise-
ments of hair-restorers. And so on and so on. It is to be noted
that not only substantives and adjectives, but adverbs and pre-
positions, may on occasion be used as complete sentences.

I think the elementary uses of a word may be distinguished as
indicative, imperative, and interrogative. When a child sees his
mother coming, he may say “mother!”; this is the indicative use.
When he wants her, he calls “mother!”; this is the imperative
use. When she dresses up as a witch and he begins to pierce the
disguise, he may say “mother?”; this is the interrogative use.
The indicative use must come first in the acquisition of language,
since the association of word and object signified can only be
created by the simultaneous presence of both. But the imperative
use very quickly follows. This is relevant in considering what
we mean by “thinking of” an object. It is obvious that the child
who has just learnt to call his mother has found verbal expression
for a state in which he had often been previously, that this state
was associated with his mother, and that it has now become
associated with the word “mother”. Before language, his state
was only partially communicable; an adult, hearing him cry,
could know that he wanted something, but had to guess what it
was. But the fact that the word “mother!” expresses his state
shows that, even before the acquisition of language, his state
had a relation to his mother, namely the relation called “thinking
of”. This relation is not created by language, but antedates it.
What language does is to make it communicable.

“Meaning” is a word which must be interpreted somewhat
differently according as it is applied to the indicative or the
imperative. In the indicative, a word A means a feature B of the
environment if, (i) when B is emphatically present to attention,
A is uttered, or there is an impulse to utter A, and (2) when A
is heard it arouses what may be called the “idea” of B, which
shows itself either in looking for B or in behaviour such as would
be caused by the presence of B. Thus in the indicative a word
“means” an object if the sensible presence of the object causes
the utterance of the word, and the hearing of the word has effects
analogous, in certain respects, to the sensible presence of the
object.

The imperative use of a word must be distinguished according
as it is heard or uttered. Broadly speaking, an imperative heard

human knowledge: its scope and limits

— e.g. the word of command in the army — is understood when
it causes a certain kind of bodily movement, or an impulse towards
such a movement. An imperative uttered expresses a desire, and
therefore requires the existence of an “idea” of the intended
effect. Thus while it “expresses” something in the speaker, it
“means” the external effect which it commands. The distinction
between what is “meant” and what is “expressed” is essential
in this use of words.

We have been concerned, in this chapter, only with the most
primitive uses of the most primitive words. We have not considered
the use of words in narrative or in hypothesis or in fiction, nor
have we examined logical words such as “not”, “or”, “all”, and
“some”; we have not inquired how learners acquire the correct
use of such words as “than” or “of”, which do not denote
recognizable features of any sensible environment. What we
have decided is that a word may become associated with some
notable feature of the environment (in general, one that occurs
frequently), and that, when it is so associated, it is also associated
with something that may be called the “idea” or “thought” of
this feature. When such an association exists, the word “means”
this feature of the environment; its utterance can be caused by
the feature in question, and the hearing of it can cause the “idea”
of this feature. This is the simplest kind of “meaning”, out of
which other kinds are developed.

Chapter III
PROPER NAMES

T here is a traditional distinction between “proper” names
and “class” names, which is explained as consisting in
the fact that a proper name applies, essentially, to only
one object, whereas a class name applies to all objects of a certain
kind, however numerous they may be. Thus “Napoleon” is a
proper name, while “man” is a class name. It will be observed
that a proper name is meaningless unless there is an object of
which it is the name, but a class name is not subject to any such
limitation. “Men whose heads do grow beneath their shoulders”
is a perfectly good class name, although there are no instances
of it. Again, it may happen that there is only one instance of a
class name, e.g. “satellite of the earth”. In such a case, the one
member may have a proper name (“the moon”), but the proper
name does not have the same meaning as the class name, and has
different syntactical functions. E.g. we can say: “ ‘Satellite of the
earth’ is a unit class”, but we cannot say “the moon is a unit
class”, because it is not a class, or at any rate not a class of the
same logical type as “satellite of the earth”, and if taken as a
class (e.g. of molecules) it is many, not one.

Many difficult questions arise in connection with proper names
Of these there are two that are especially important: first, what
is the precise definition of proper names? second, is it possible
to express all our empirical knowledge in a language containing
no proper names? This second question, we shall find, takes us
to the heart of some of the most ancient and stubborn of philo-
sophical disputes.

In seeking a definition of “proper name”, we may approach
the subject from the point of view of metaphysics, logic, physics,
syntax, or theory of knowledge. I will say a few preliminary words
about each of these.

A. Metaphysical — It is fairly obvious that proper names owe
their existence in ordinary language to the concept of “substance”
— originally in the elementary form of “persons” and “things”.
A substance or entity is named, and then properties are assigned
to it. So long as this metaphysic was accepted, there was no

human knowledge: its scope and limits

difficulty as to proper names, which were the designations of such
substances as were sufficiently interesting. Sometimes, it is true,
we should give a name to a collection of substances, such as
France or the sun. But such names, strictly speaking, were not
necessary. In any case, we could extend our definition to embrace
collections of substances.

But most of us, nowadays, do not accept “substance” as a
useful notion. Are we then to adopt, in philosophy, a language
without proper names? Or are we to find a definition of “proper
name” which does not depend on “substance”? Or are we to
conclude that the conception of “substance” has been too hastily
rejected? For the present, I merely raise these questions, without
attempting to answer them. All that I want to make clear at the
moment is that proper names, as ordinarily understood, are ghosts
of substances.

B. Syntactical . — It i9 clear that a syntactical definition of
“proper name” must be relative to a given language or set of
languages. In the languages of daily life, and also in most of those
employed in logic, there is a distinction between subject and
predicate, between relation-words and term-words. A “name” will
be, in such languages, “a word which can never occur in a sentence
except as a subject or a term-word”. Or again: a proper name
is a word which may occur in any form of sentence not containing
variables, whereas other words can only occur in sentences of
appropriate form. Sometimes it is said that some words are
“syncategorimatic”, which apparently means that they have no
significance by themselves, but contribute to the significance of
sentences in which they occur. According to this way of speaking,
proper names are not syncategorimatic, but whether this can be
a definition is a somewhat doubtful question. In any case, it is
difficult to get a clear definition of the term “syncategorimatic”.

The chief inadequacy of the above syntactical point of view is
that it does not, in itself, help us to decide whether it is possible
to construct languages with a different kind of syntax, in which
the distinctions we have been considering would disappear.

C. Logical . — Pure logic has no occasion for names, since its
propositions contain only variables. But the logician may wonder,
in his unprofessional moments, what constants could be sub-
stituted for his variables. The logician announces, as one of his
principles, that, if “fx” is true for every value of “#”, then "/»”

PROPER NAMES

is true, where “a” is any constant. This principle does not mention
a constant, because “any constant” is a variable ; but it is intended
to justify those who want to apply logic. Every application of logic
or mathematics consists in the substitution of constants for
variables; it is therefore essential, if logic or mathematics is to
be applied, to know what sort of constants can be substituted
for what sort of variables. If any kind of hierarchy is admitted
among variables, “proper names” will be “constants which are
values of variables of lowest type”. There are, however, a number
of difficulties in such a view. I shall not therefore pursue it further.

D. Physical . — There are here two points of view to be con-
sidered. The first is that a proper name is a word designating any
continuous portion of space-time which sufficiently interests us;
the second is that, this being the function of proper names, they
are unnecessary, since any portion of space-time can be described
by its co-ordinates. Carnap {Logical Syntax , pp. 12-13) explains
that latitude and longitude, or space- time co-ordinates, can be
substituted for place-names. “The method of designation by
proper names is the primitive one ; that of positional designation
corresponds to a more advanced stage of science, and has con-
siderable methodological advantages over the former.” In the
language he employs, co-ordinates, he says, replace such words
as “Napoleon” or “Vienna”. This point of view deserves full
discussion, which I shall undertake shortly.

E. Epistemological . — We have here, first, a distinction not
identical with that between proper names and other words, but
having perhaps some connection with it. This is the distinction
between words having a verbal definition and words having only
an ostensive definition. As to the latter, two points are obvious:
(1) not all words can have verbal definitions; (2) it is largely
arbitrary which words are to have only ostensive definitions.
E.g. if “Napoleon” is defined ostensively, “Joseph Bonaparte”
may be defined verbally as “Napoleon's oldest brother”. However,
this arbitrariness is limited by the fact that, in the language of
a given person, ostensive definitions are only possible within the
limits of his experience. Napoleon’s friends might (subject to
limitations) define him ostensively, but we cannot, since we can
never say truly “ that is Napoleon”. There is obviously here a
problem connected with that of proper names; how closely, I
shall not discuss at present.

HUMAN knowledge: its scope and limits

We have, it is clear, a number of problems to consider, and,
as is apt to happen in philosophy, it is difficult to be clear as to
what precisely the problems are. I think we shall do best if we
begin with Carnap’s substitution of co-ordinates for proper names.
The question we have to consider is whether such a language
can express the whole of our empirical knowledge.

In Carnap’s system, a group of four numbers is substituted for
a space-time point. He illustrates by the example “Blue x ly x 2 ,
# 3 , jc 4 , meaning “the position (x v x 2f x 3l x 4 ) is blue”, instead of
“Blue (a)” meaning “the object a is blue”. But now consider
such a sentence as “Napoleon was in Elba during part of 1814”.
Carnap, I am sure, will agree that this sentence is true, and that
its truth is empirical, not logical. But if we translate it into his
language it will become a logical truth. “Napoleon” will be
replaced by “all quartets of numbers enclosed within such-and-
such boundaries”; so will “Elba”, and so will “1814”. We shall
then be stating that these three classes of quartets have a common
part. This, however, is a fact of logic. Clearly this is not what
we meant. We give the name “Napoleon” to a certain region, not
because we are concerned with topology, but because that region
has certain characteristics which make it interesting. We may
defend Carnap by supposing, adopting a schematic simplification,
that “Napoleon” is to mean “all regions having a certain quality
say N”, while “Elba” is to mean “all regions having the quality E”.
Then “Napoleon spent some time in Elba” will become: “The
regions having the quality N and those having the quality E
overlap.” This is no longer a fact of logic. But it has interpreted
the proper names of ordinary language as disguised predicates.

But our schematic simplification is too violent. There is no
quality, or collection of qualities, present wherever Napoleon was
and absent wherever he was not. As an infant, he did not wear
a cocked hat, or command armies, or fold his arms, while all these
things were also done, at times, by other people. How, then, are
we to define the word “Napoleon”? Let us continue to do our
best for Carnap. In the moment of baptism, the priest decides
that the name “Napoleon” is to apply to a certain small region
in his neighbourhood, which has a more or less human shape,
and that it is to apply to other future regions connected with this
one, not only by continuity, which is not sufficient to secure
material identity, but by certain causal laws, those, namely, which

PROPER NAMES

lead us to regard a body on two occasions as that of the same
person. We may say: Given a temporally brief region having the
characteristics of a living human body, it is an empirical fact that
there are earlier and later regions connected with this one by
physical laws, and having more or less similar characteristics ; the
total of such regions is what we call a “person”, and one such
region was called “Napoleon”. That naming is retro-active appears
from a plaque on a certain house in Ajaccio saying: “Ici Napoleon
fut con9u.”

This may be accepted as an answer to the objection that, on
Carnap’s view, “Napoleon was once in Elba” would be a pro-
position of logic. It leaves, however, some very serious questions.
We saw that “Napoleon” cannot be defined simply by qualities,
unless we are to hold it impossible that there should be two
exactly similar individuals. One of the uses of space-time, how-
ever, is to differentiate similar individuals in different places.
Carnap has his sentences “Blue (3)”, “Blue (4)”, etc., meaning
“the place 3 is blue”, “the place 4 is blue”, etc. We can, it is
supposed, distinguish blue in one place from blue in another.
But how are the places distinguished? Carnap takes space-time
for granted, and never discusses how space-time places are dif-
ferentiated. In fact, in his system, space-time regions have the
characteristics of substance. The homogeneity of space-time is
assumed in physics, and yet it is also assumed that there are
different regions, which can be distinguished. Unless we are to
accept the objectionable metaphysics of substance, wc shall have
to suppose the regions distinguished by differences of quality.
We shall then find that the regions need no longer be regarded
as substantial, but as bundles of qualities.

Carnap’s co-ordinates, which replace names, are, of course, not
assigned quite arbitrarily. The origin and the axes are arbitrary,
but when they are fixed, the rest proceeds on a plan. The year
which we call “1814” is differently named by the Mohammedans,
who date from the Hejira, and by the Jewish era, which dates
from the Creation. But the year we call “1815” will have the
next number, in any system, to that given to what we call “1814”.
It is because co-ordinates are not arbitrary that they are not
names. Co-ordinates describe a point by its relations to the origin
and the axes. But we must be able to say “ this is the origin”. If
we are to be able to say this, we must be able to name the origin,

HUMAN knowledge: its scope and limits

or to describe it in some way, and at first sight it might be thought
that any possible way would be found to involve names. Take,
for instance, longitude. The origin of longitude is the meridian
of Greenwich, but it might equally well be any other meridian.
We cannot define “Greenwich” as “longitude o°, latitude 52 0 ”,
because, if we do, there is no means of ascertaining where longi-
tude o° is. When we say “longitude o° is the longitude of Green-
wich”, what we say is satisfactory because we can go to Greenwich
and say “this is Greenwich”. Similarly, if we live at (say) longi-
tude 40° W., we can say “the longitude of this place is 40° W.”,
and then we can define longitude o° by relation to this place. But
unless we have a way of knowing some places otherwise than by
latitude and longitude, latitude and longitude become unmeaning.
When we ask “what are the latitude and longitude of New York?”
we are not asking the same sort of question as we should be if
we descended into New York by a parachute and asked “what
is the name of this city?” We are asking: “How far is New York
west of Greenwich and north of the equator?” This question
supposes New York and Greenwich known and already named.

It would be possible to assign a finite number of co-ordinates
at haphazard, and then they would all be names. When (as is
always done) they are assigned on a principle, they are descrip-
tions, defining points by their relations to the origin and the axes.
But these descriptions fail for the origin and the axes themselves,
since, as regards them, the numbers are assigned arbitrarily. To
answer the question “where is the origin?” we must have some
method of identifying a place without mentioning its co-ordinates.
It is the existence of such methods that is presupposed by the
use of proper names.

I conclude, for the moment, that we cannot wholly dispense
with proper names by means of co-ordinates. We can perhaps
reduce the number of proper names, but we cannot avoid them
altogether. Without proper names we can express the whole of
theoretical physics, but no part of history or geography; this, at
least, is our provisional conclusion so far, but we shall find reason
to modify it later.

Let us consider a little further the substitution of descriptions
for names. Somebody must be the tallest man now living in the
United States. Let us suppose he is Mr. A. We may then, in
place of “Mr. A”, substitute “the tallest man now living in the

PROPER NAMES

United States”, and this substitution will not, as a rule, alter the
truth or falsehood of any sentence in which it is made. But it
will alter the statement. One may know things about Mr. A that
one does not know about the tallest man in the United States,
and vice versa. One may know that Mr. A lives in Iowa, but not
that the tallest man in the United States lives in Iowa. One may
know that the tallest man in the United States is over ten years
old, but one may not know whether Mr. A is man or boy. Then
there is the proposition “Mr. A is the tallest man in the United
States.” Mr. A may not know this; there may be a Mr. B who
runs him close. But Mr. A certainly knows that Mr. A is Mr. A.
This illustrates once more that there are some things which cannot
be expressed by means of descriptions substituted for names.

The names of persons have verbal definitions in terms of “this”.
Suppose you are in Moscow and some one says “that’s Stalin”,
then “Stalin” is defined as “the person whom you are now seeing”
— or, more fully: “that series of occurrences, constituting a
person, of which this is one”. Here “this” is undefined, but
“Stalin” is defined. I think it will be found that every name
applied to some portion of space-time can have a verbal definition
in which the word “this”, or some equivalent, occurs. This, I
should say, is what distinguishes the name of an historical character
from that of an imaginary person, such as Hamlet. Let us take
a person with whom we are not acquainted, say Socrates. We
may define him as “the philosopher who drank the hemlock”,
but such a definition does not assure us that Socrates existed,
and if he did not exist, “Socrates” is not a name. What does
assure us that Socrates existed ? A variety of sentences heard or
read. Each of these is a sensible occurrence in our own experience.
Suppose we find in the Encyclopaedia the statement “Socrates
was an Athenian philosopher”. The sentence, while we see it,
is a this , and our faith in the Encyclopaedia leads us to say “this
is true”. We can define “Socrates” as “the person described in
the Encyclopaedia under the name ‘Socrates’ ”. Here the name
“Socrates” is experienced. We can of course define “Hamlet” in
a similar way, but some of the propositions used in the definition
will be false. E.g. if we say “Hamlet was a Prince of Denmark
who was the hero of one of Shakespeare’s tragedies”, this is false.
What is true is: “ ‘Hamlet’ is a word which Shakespeare pretends
to be the name of a Prince of Denmark”. It would thus seem to

HUMAN knowledge: its scope and limits

follow that, apart from such words as “this” and “that”, every
name is a description involving some this , and is only a name
in virtue of the truth of some proposition. (The proposition may
be only “this is a name”, which is false if this is “Hamlet”.)

We must consider the question of minimum vocabularies. I call
a vocabulary a “minimum” one if it contains no word which is
capable of a verbal definition in terms of the other words of the
vocabulary.^ Two minimum vocabularies dealing with the same
subject-matter may not be equal ; there may be different methods
of definition, some of which lead to a shorter residuum of
undefined terms than others do. The question of minimum
vocabularies is sometimes very important. Peano reduced the
vocabulary of arithmetic to three words. It was an achievement in
classical physics when all units were defined in terms of the units
of mass, length, and time. The question I wish to discuss is:
What characteristics must belong to a minimum vocabulary by
means of which we can define all the words used in expressing
our empirical knowledge or beliefs, in so far as such words have
any precise meaning? More narrowly, to revert to a former
example, what sort of minimum vocabulary is needed for
“Napoleon was in Elba during part of 1814” and kindred state-
ments? Perhaps, when we have answered this, we shall be able
to define “names”. I assume, in the following discussion, that
such historical-geographical statements are not analytic, that is
to say, though they are true as a matter of fact, it would not be
logically impossible for them to be false.

Let us revert to the theory, which is suggested by what Carnap
says, that “Napoleon” is to be defined as a certain region of
space-time. We objected that, in that case, “Napoleon was for
a time in Elba” is analytic. It may be retorted: yes, but to find
out what is not analytic you must inquire why we give a name
to the portion of space-time that was Napoleon. We do so because
it had certain peculiar characteristics. It was a person, and when
adult it wore a cocked hat. We shall then say: “This portion of
space-time is a person, and in its later portions it wears a cocked
hat; that portion of space- time is a small island; this and that
have a common part”. We have here three statements, the first
two empirical, the third analytic. This seems unobjectionable.
It leaves us with the problem of assigning co-ordinates, and also
with that of defining such terms as “person” and “island”. Such

PROPER NAMES

terms as “person” and “island” can obviously be defined in terms
of qualities and relations; they are general terms, and not (one
would say) such as lead to proper names. The assigning of co-
ordinates requires the assigning of origin and axes. We may, for
simplicity, ignore the axes and concentrate on the origin. Can
the origin be defined ?

Suppose, for example, you are engaged in planetary theory, not
merely in a theoretical spirit, but with a view to the testing of
your calculations by observations. Your origin, in that case, will
have to be defined by something observable. It is universally
agreed that absolute physical space-time is not observable. The
things we can observe are, broadly speaking, qualities and spatio-
temporal relations. We can say “I shall take the centre of the
sun as my origin”. The centre of the sun is not observable, but
the sun (in a sense) is. It is an empirical fact that I frequently
have an experience which I call “seeing the sun”, and that I can
observe what seem to be other people having a similar experience.
“The sun” is a term which can be defined by qualities: round,
hot, bright, of such-and-such apparent size, etc. It happens that
there is only one object in my experience having these qualities,
and that this object persists. I can give it a proper name, “the
sun”, and say “I shall take the sun as my origin”. But since I
have defined the sun by its qualities, it does not form part of
a minimum vocabulary. It seems to follow that, while the words
for qualities and spatio-temporal relations may form part of my
minimum vocabulary, no words for physical spatio-temporal
regions can do so. This is, in fact, merely a way of stating that
physical spatio-temporal position is relative, not absolute.

Assuming this correct so far, the question arises whether we
need names for qualities and spatio-temporal relations. Take
colours, for example. It may be said that they can be designated
by wave-lengths. This leads to Carnap’s contention that there is
nothing in physics which cannot be known to a blind man. So
far as theoretical physics is concerned, this is obviously true. It
is true also, up to a point, in the empirical field. We see that the
sky is blue, but a race of blind men could devise experiments
showing that transverse waves of certain wave-lengths proceed
from it, and this is just what the ordinary physicist qud physicist,
is concerned to assert. The physicist, however, does not trouble
to assert, and the blind man cannot assert, the proposition : “When

human knowledge: its scope and limits

light of a certain frequency strikes a normal eye, it causes a
sensation of blue. ,, This statement is not a tautology; it was a
discovery, made many thousands of years after words for “blue”
had been in common use.

The question whether the word “blue” can be defined is not
easy. We might say: “Blue” is the name of colour-sensations
caused by light of such-and-such frequencies. Or we might say:
“Blue” is the name of those shades of colour which, in the spectrum,
come between violet and green. Either of these definitions might
enable us to procure for ourselves a sensation of blue. But when
we had done so, we should be in a position to say: “So that is
blue .” This would be a discovery, only to be made by actually
experiencing blue. And in this statement, I should say, “that”
is in one sense a proper name, though of that peculiar sort that
I call “egocentric”.

We do not usually give names to smells and tastes, but we
could do so. Before going to America, I knew the proposition “the
smell of a skunk is disagreeable”. Now I know the two proposi-
tions: “ that is the smell of a skunk”, and “ that is disagreeable”.
Instead of “that”, we might use a name, say “pfui”, and should
do so if we often wished to speak of the smell without men-
tioning skunks. But to any one who had not had the requisite
experience, the name would be an abbreviated description, not
a name.

I conclude that names are to be applied to what is experienced,
and that what is experienced does not have, essentially and neces-
sarily, any such spatio-temporal uniqueness as belongs to a space-
time region in physics. A word must denote something that can
be recognized , and space-time regions, apart from qualities, cannot
be recognized, since they are all alike. They are in fact logical
fictions, but I am ignoring this for the moment.

There are occurrences that I experience, and I believe there
are others that I do not experience. The occurrences that I
experience are all complex, and can be analysed into qualities
with spatial and temporal relations. The most important of these
relations are compresence, contiguity, and succession. The words
that we use to designate qualities are not precise; they all have
the sort of vagueness that belongs to such words as “bald” and
“fat”. This is true even of the words that we are most anxious
to make precise, such as “centimetre” and “second”. Words

PROPER NAMES

designating qualities must be defined ostensively, if we are to be
able to express observations; as soon as we substitute a verbal
definition, we cease to express what is observed. The word “blue,”
for instance, will mean “a colour like that ”, where that is a blue
patch. How like that it must be to be blue, we cannot state with
any precision.

This is all very well, but how about such words as “this” and
“that”, which keep on intruding themselves? We think of the
word “this” as designating something which is unique, and can
only occur once. If, however, “this” denotes a bundle of corn-
present qualities, there is no logical reason why it should not
recur. I accept this. That is to say, I hold that there is no
class of empirically known objects such that, if x is a mem-
ber of the class, the statement “ x precedes x ” is logically
impossible.

We are accustomed to think that the relation “precedes” is
asymmetrical and transitive . 1 “Time” and “event” are both con-
cepts invented with a view to securing these properties to the
relation “precedes”. Most people have discarded “time” as some-
thing distinct from temporal succession, but they have not dis-
carded “event”. An “event” is supposed to occupy some con-
tinuous portion of space- time, at the end of which it ceases, and
cannot recur. It is clear that a quality, or a complex of qualities,
may recur; therefore an “event”, if non-recurrence is logically
necessary, is not a bundle of qualities. What, then, is it, and how
is it known? It will have the traditional characteristics of sub-
stance, in that it will be a subject of qualities, but not defined
when all its qualities are assigned. And how do we know that
there is any class of objects the members of which cannot recur?
If we are to know this, it might seem that it must be a case of
synthetic a priori knowledge, and that if we reject the synthetic
a priori , we must reject the impossibility of recurrence. We shall,
of course, admit that, if we take a sufficiently large bundle of
qualities, there will be no empirical instance of recurrence. Non-
recurrence of such bundles may be accepted as a law of physics,
but not as something necessary.

The view that I am suggesting is that an “event” may be
defined as a complete bundle of compresent qualities, i.e. a bundle

1 I.e. that if A precedes B, B does not precede A, and if A precedes B
and B precedes C, then A precedes C.

97 G

human knowledge: its scope and limits

having the two properties (a) that all the qualities in the bundle
are compresent, (6) that nothing outside the bundle is compresent
with every member of the bundle. I assume that, as a matter of
empirical fact, no event recurs; that is to say, if a and b are
events, and a is earlier than b y there is some qualitative difference
between a and b . For preferring this theory to one which makes
an event indefinable, there are all the reasons commonly alleged
against substance. If two events were exactly alike, nothing could
ever lead us to suppose that they were two. In taking a census,
we could not count one apart from the other, for, if we did, that
would be a difference between them. And from the standpoint
of language, a word must denote something that can be recog-
nized, and this requires some recognizable quality. This leads
to the conclusion that words such as “Napoleon” can be defined,
and are therefore theoretically unnecessary; and that the same
thing would be true of words designating events, if we were
tempted to invent such words.

I conclude that, if we reduce our empirical vocabulary to a
minimum, thereby excluding all words that have verbal definitions,
we shall still need words for qualities, compresence, succession,
and observed spatial relations, i.e. spatial relations which can be
discriminated within a single sensible complex. It is an empirical
fact that, if we form a complex of all the qualities that are all
compresent with each other, this complex is found, so far as our
experience goes, not to precede itself, i.e. not to recur. In forming
the time-series, we generalize this observed fact.

The nearest approach to proper names in such a language
will be the words for qualities and complexes of compresent
qualities. These words will have the syntactical characteristics of
proper names, but not certain other characteristics that we expect,
for example that of designating a region which is spatio-tem-
porally continuous. Whether, in these circumstances, such words
are to be called “names”, is a matter of taste, as to which I
express no opinion. What are commonly called proper names —
e.g. “Socrates” — can, if I am right, be defined in terms of qualities
and spatio-temporal relations, and this definition is an actual
analysis. Most subject-predicate propositions, such as “Socrates
is snub-nosed”, assert that a certain quality, named by the
predicate, is one of a bundle of qualities named by the subject —
this bundle being a unity in virtue of compresence and causal

PROPER NAMES

relations. Proper names in the ordinary sense, if this is right, are
misleading, and embody a false metaphysic.

Note . — The above discussion of proper names is not intended
to be conclusive. The subject will be resumed in other contexts,
especially in Part IV, Chapter VIII.

Chapter IV

EGOCENTRIC PARTICULARS

I give the name “egocentric particulars” to words of which
the meaning varies with the speaker and his position in time
and space. The four fundamental words of this sort are “I”,
“this”, “here”, and “now”. The word “now” denotes a different
point of time on each successive occasion when I use it; the word
“here” denotes a different region of space after each time when
I move; the word “I” denotes a different person according to
who it is that utters it. Nevertheless there is obviously some sense
in which these words have a constant meaning, which is the reason
for the use of the words. This raises a problem, but before con-
sidering it let us consider what other words are egocentric, and
especially what words are really egocentric although intended
not to be so.

Among obviously egocentric words are “near” and “far”,
“past”, “present”, and “future”, “was”, “is”, and “will be”, and
generally all forms of verbs involving tense. “This” and “that”
are obviously egocentric; in fact, “this” might be taken as the
only egocentric word not having a nominal definition. We could
say that “I” means “the person experiencing this”, “now” means
“the time ol this”, and “here” means “the place of this”. The
word “this” is, in a sense, a proper name, but it differs from true
proper names in the fact that its meaning is continually changing.
This does not mean that it is ambiguous, like (say) “John Jones”,
which is at all times the proper name of many different men.
Unlike “John Jones”, “this” is at each moment the name of only
one object in one person’s speech. Given the speaker and the
time, the meaning of “this” is unambiguous, but when the speaker
and the time are unknown we cannot tell what object it denotes.
For this reason, the word is more satisfactory in speech than in
print. If you hear a man say “this is an age of progress”, you
know what age he refers to ; but if you read the same statement
in a book it may be what Adam said when he invented the spade
or what was said by any later optimist. You can only decide what
the statement means by finding out when it was written, and in

ioo

EGOCENTRIC PARTICULARS

this sense its meaning is not self-contained but requires elucidation
by extraneous information.

One of the aims of both science and common sense is to replace
the shifting subjectivity of egocentric particulars by neutral public
terms. “I” is replaced by my name, “here” by latitude and
longitude, and “now” by date. Suppose I am walking with a
friend on a dark night, and we lose touch with each other: he
calls out “where are you?” and I reply “here I am”. Science will
not accept such language; it will substitute “At 11.32 p.m. on
January 30, 1946, B.R. was at longitude 4 0 3' 29" W. and at
latitude 53 0 16' 14" N”. This information is impersonal: it gives
a prescription by which a qualified person who possesses a sextant
and a chronometer, and has the patience to wait for a sunny day,
can determine where I was, which he may proclaim in the words
“here is where he was”. If the matter is of sufficient importance,
say in a trial for murder, this elaborate procedure may be worth
the trouble it involves. But its appearance of complete imper-
sonality is in part deceptive. Four items are involved: my name,
the date, latitude, and longitude. In regard to each of these there
is an element of egocentricity which is concealed by the fact that,
for most purposes, it has no practical importance.

From a practical point of view, the impersonality is complete.
Two competent persons, given time and opportunity, will both
accept or both reject a statement of the form: “At time t y A was
at longitude B, latitude C.” Let us call this statement “P”. There
is a procedure for determining date, latitude, and longitude, which,
if correctly observed, leads different people to the same result,
in the sense that, if both say truly “he was here five minutes
ago”, they must be in each other’s presence. This is the essential
merit of scientific terminology and scientific technique. But when
we examine closely the meanings of our scientific terms we find
that the subjectivity we sought to avoid has not been wholly
banished.

Let us begin with my name. We substitute “B.R.” for “I” or
“you” or “he”, as the case may be, because “B.R.” is a public
appellation, appearing on my passport and my identity card. If
a policeman says “who are you?” I might reply by saying “look I
this is who I am”, but this information is not what the policeman
wants, so I produce my identity card and he is satisfied. But
essentially I have only substituted one sensible impression for

101

human knowledge: its scope and limits

)

another. In looking at the identity card the policeman acquires
a certain visual impression, which enables him to say “the name
of the accused is B.R.” Another policeman, looking at the same
identity card, will utter what is called the “same” sentence, that
is to say, he will emit a series of noises closely similar to those
emitted by the first policeman. It is this similarity, mistakenly
regarded as identity, which is the merit of the name. If the two
policemen had had to describe my appearance, the first, delaying
me at the end of an all-day walk in the rain, might say “he was
a furious red-faced tramp”, while the other might say “he was
a benign old gentleman in evening dress”. The name has the
merit of being less variable, but it remains something known only
through the sensible impressions of individuals, of which no two
are exactly alike. We always come back to “this is his name”,
where this is a present occurrence. Or rather, to be exact, “his
name is a class of sensible occurrences all very similar to this”.
We secure, by our procedure, a method of providing sets of
closely similar occurrences, but we do not wholly escape from
“this”.

There is involved here a principle of considerable scope and
importance, which deserves a more detailed exposition, to which
we must now devote ourselves.

Let us begin with a homely illustration. Suppose you are
acquainted with a certain Mrs. A, and you know that her mother,
whom you have never met, is called Mrs. B. What does the name
“Mrs. B” mean for you? Not what it means for those who know
her, still less what it means for her herself. It must mean some-
thing definable in terms of your experience, as must every word
that you can use understandingly. For every word that you can
understand must either have a nominal definition in terms of
words having ostensive definitions, or must itself have an osten-
sive definition; and ostensive definitions, as appears from the
process by which they are effected, are only possible in relation
to events that have occurred to you. Now the name “Mrs. B”
is something that you have experienced; therefore when you speak
of Mrs. B you may be mentally defining her as “the lady whose
name is ‘Mrs. B’ ”. Or, if one were to concede (what would not
be strictly accurate) that you are acquainted with Mrs. A, you
might define “Mrs. B” as “the mother of Mrs. A”. In this way,
although Mrs. B is outside your experience, you can interpret

102

EGOCENTRIC PARTICULARS

sentences in which her name occurs in such a way that your lack
of experience does not prevent you from knowing if the sentences
are true.

We can now generalize the process involved in the above illus-
tration. Suppose there is some object a which you know by
experience, and suppose you know (no matter how) that there is
just one object to which a has a known relation R, but there is
no such object in your experience. (In the above case, a is a
Mrs. A, and R is the relation of daughter to mother.) You can
then give a name to the object to which a has the relation R;
let the name be “6”. (In our illustration it was “Mrs. B”.) It
then becomes easy to forget that b is unknown to you although
you may know multitudes of true sentences about b. But in fact,
to speak correctly, you do not know sentences about b ; you know
sentences in which the name u b" is replaced by the phrase “the
object to which a has the relation R”. You know also that there
are sentences about the actual object b which are verbally iden-
tical with those that you know about the object to which a has
the relation R — sentences pronounced by other people in which
occurs as a name — but although you can describe these
sentences, and know (within common-sense limits) which are true
and which false, you do not know the sentences themselves. You
may know that Mrs. A’s mother is rich, but you do not know
what Mrs. B knows when she says “I am rich”.

The result of this state of affairs is that our knowledge seems
to extend much further beyond our experience than it actually
does. We may perhaps distinguish, in such cases as we have been
considering, between what we can assert and what we intend.
If I say “Mrs. B is rich”, I intend something about Mrs. B her-
self, but what I actually assert is that Mrs. A has a rich mother.
Another person may know of Mrs. B, not as the mother of Mrs. A,
but as the mother of another daughter Mrs. C. In that case, when
he says “Mrs. B is rich”, he means “Mrs. C has a rich mother”,
which is not what I meant. But we both intend to say something
about Mrs. B herself, though in this neither of us is successful.
This does not matter in practice, as the things we respectively
say about Mrs. A’s mother or Mrs. C’s mother would be true
of Mrs. B if only we could say them. But although it does not
matter in practice, it matters greatly in theory of knowledge. For
in fact everybody except myself is to me in the position of Mrs. B ;

103

HUMAN knowledge: its scope and limits

of our supposed subjective prison. In this process of escape the
interpretation of egocentric particulars is a very essential step.

Before attempting a precise account of egocentric words, let us
briefly survey the picture of the world to which subsequent
discussions will lead us.

There is one public space, namely the space of physics, and
this space is occupied by public physical objects. But public
space and public objects are not sensible; they are arrived at by
a mixture of inference and logical construction. Sensible spaces
and sensible objects differ from one person to another, though
they have certain affinities to each other and to their public
counter-parts.

There is one public time , 1 in which not only physical events,
but mental events also, have their place. There are also private
times, which are those given in memory and expectation.

My whole private space is “here” in physical space, and my
whole private time is “now” in public time. But there are also
private “heres” and “nows” in private spaces and times.

When your friend calls out in the dark “where are you?” and
you answer “here I am”, the “here” is one in physical space,
since you are concerned to give information which will help
another to find you. But if, when alone, you are looking for a lost
object, and on finding it you exclaim “here it is”, the “here” may
be either in public space or in your private space. Of course
ordinary speech does not distinguish between public and private
space. Broadly, “here” is where my body is — my physical body if
I mean “here” in physical space, and my percept of my body if
I mean “here” in my private space. But “here” may be much
more narrowly localized, for instance if you are pointing out a
thorn in your finger. One might say (though this would not quite
accord with usage) that “here” is the place of whatever sensible
object is occupying my attention. This, though not quite the
usual meaning of the word, is the concept which most needs
discussing in connection with the word “here”.

“Now” has a similar two-fold meaning, one subjective and one
objective. When I review my life in memory, some of the things

1 This is subject to limitations connected with relativity. But as lan-
guage and theory of knowledge are concerned with inhabitants of the
earth, these may be ignored, since no two people have a relative velocity
comparable to that of light.

106

EGOCENTRIC PARTICULARS

I remember seem a long time ago, others more recent, but all
are in the past as compared with present percepts. This “past-
ness”, however, is subjective: what I am remembering, I re-
member now , and my recollecting is a present fact. If my memory
is veridical, there was a fact to which my recollection has a certain
relation, partly causal, partly of similarity ; this fact was objectively
in the past. I maintain that, in addition to the objective relation
of before-and-after, by which events are ordered in a public
time-series, there is a subjective relation of more-or-less-remote,
which holds between memories that all exist at the same objective
time. The private time-series generated by this relation differs
not only from person to person, but from moment to moment in
the life of any one person. There is also a future in the private
time-series, which is that of expectation. Both private and public
time have, at each moment in the life of a percipient, one peculiar
point, which is, at that moment, called “now”.

It is to be observed that “here” and “now” depend upon
perception; in a purely material universe there would be no
“here” and “now”. Perception is not impartial, but proceeds
from a centre; our perceptual world is (so to speak) a perspective
view of the common world. What is near in time and space
generally gives rise to a more vivid and distinct memory or percept
than what is far. The public world of physics has no such centre
of illumination.

In defining egocentric particulars, we may take “this” as
fundamental, in a sense in which “this” is not distinguished from
“that”. I shall attempt an ostensive definition of “this”, and
thence a nominal definition of the other egocentric particulars.

^This” denotes whatever, at the moment when the word is
used, occupies the centre of attention. With words which are not
egocentric, what is constant is something about the object
indicated, but “this” denotes a different object on each occasion
of its use: what is constant is not the object denoted, but its
relation to the particular use of the word. Whenever the word is
used, the person using it is attending to something, and the word
indicates this something. When a word is not egocentric, there is
no need to distinguish between different occasions when it is
used, but we must make this distinction with egocentric? words,
since what they indicate is something having a given relation to
the particular use of the word. ^

107

HUMAN knowledge: its scope and limits

We may define “I” as “the person attending to this”, “now”
as “the time of attending to this”, and “here” as “the place of
attending to this”. We could equally well take “here-now” as
fundamental; then “this” would be defined as “what is here-
now”, and “I” as “what experiences this”.

Can two persons experience the same “this”, and if so, in what
circumstances? I do not think this question can be decided by
logical considerations : a priori , either answer would be possible.
But taking the question empirically, it has an answer. When the
“this” concerned is what common sense takes to be a percept of
a physical object, difference of perspective makes a difference in
the percept unavoidable, if the same physical object is concerned
in the two cases. Two people looking at one tree, or listening to
the song of one bird, are having somewhat different percepts.
But two people looking at different trees might, theoretically, have
exactly similar percepts, though this would be improbable. Two
people may see exactly the same shade of colour, and are likely
to do so if each is looking at a continuous band of colours, e.g.
those of the rainbow. Two people looking at a square table will
not see exactly similar quadrilaterals, but the quadrilaterals they
see will have certain geometrical properties in common.

It thus appears that two people are more likely to have the
same “this” if it is somewhat abstract than if it is fully concrete.
In fact, broadly speaking, every increase of abstractness diminishes
the difference between one person’s world and another’s. When
we come to logic and pure mathematics, there need be no difference
whatever : two people can attach exactly the same meaning to the
word “or” or the word “371 ,294”. This is one reason why physics,
in its endeavour to eliminate the privacy of sense, has grown
progressively more abstract. This is also the reason for the view,
which has been widely held by philosophers, that all true know-
ledge is intellectual rather than sensible, and that the intellect
iberates while the senses keep us in a personal prison. In such
views there is an element of truth, but no more, except where
logic and pure mathematics are concerned; for in all empirical
knowledge liberation from sense can be only partial. It can,
however, be carried to the point where two men’s interpretations
of a given sentence are nearly certain to be both true or both false.
The securing of this result is one of the aims (more or less un-
conscious) governing the development of scientific concepts.

108

Chapter V

SUSPENDED REACTIONS: KNOWLEDGE
AND BELIEF

W E have been concerned hitherto with what may be
called the “exclamatory” use of language, when it is
used to denote some interesting feature of a man's
present experience. So long as this use alone is in question, a
single word can function as a sentence in the indicative. When
Xenophon's Ten Thousand exclaimed “Sea! Sea!” they were
using the word in this way. But a single word may also be used
in other ways. A man found dying of thirst in the desert may
murmur “water!” and is then uttering a request or expressing a
desire; he may see a mirage and say “water?”; or he may see a
spring and assert “water”. Sentences are needed to distinguish
between these various uses of words. They are needed also — and
this is perhaps their main use — to express what may be called
“suspended reactions”. Suppose you intend to take a railway
journey to-morrow, and you look up your train to-day : you do not
propose, at the moment, to take any further action on the know-
ledge you have acquired, but when the time comes you will
behave in the appropriate manner. Knowledge, in the sense in
which it does not merely register present sensible impressions,
consists essentially of preparations for such delayed reactions.
Such preparations may in all cases be called “beliefs”, but they
are only to be called “knowledge” when they prompt successful
reactions, or at any rate show themselves related to the facts with
which they are concerned in some way which distinguishes them
from preparations that would be called “errors”.

It is important not to exaggerate the role of language. In my
view, there is in pre-linguistic experience something that may be
called “belief”, and that may be true or false; there are also, I
should say, what may be called “ideas”. Language immensely
increases the number and complexity of possible beliefs and ideas,
but is not, I am convinced, necessary for the simplest beliefs and
ideas. A cat will watch for a long time at a mousehole, with her
tail swishing in savage expectation; in such a case, one should
say (so I hold) that the smell of mouse stimulates the “idea” of

109

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

the rest of what makes up an actual mouse. The objection to such
language comes, it seems to me, from an unduly intellectualist
conception of what is meant by the word “idea”. I should define
an “idea” as a state of an organism appropriate (in some sense)
to something not sensibly present. All desire involves ideas in
this sense, and desire is certainly pre-linguistic. Belief also, in an
important sense, exists in the cat watching the mousehole, belief
which is “true” if there is a mouse down the hole and “false” if
not.

The word “mouse”, by itself, will not express the different
attitudes of the cat while waiting for her prey and when seizing
it; to express these different attitudes further developments of
language are necessary. Command, desire, and narrative all
involve the use of words describing something not sensibly
present, and to distinguish them from each other and from the
indicative various linguistic devices are necessary.

Perhaps the necessity to assume “ideas” as existing ante-
cedently to language may be made more evident by considering
what it is that words express. The dying man in the desert who
murmurs “water 1” is clearly expressing a state in which a dying
animal might be. How this state should be analysed is a difficult
question, but we all, in a sense, know the meaning of the word
“thirst”, and we all know that what this word means does not
depend for its existence upon there being a word to denote it.
The word “thirst” denotes a desire for something to drink, and
such a desire involves, in the sense already explained, the presence
of the “idea” of drink. What would commonly be called a man's
“mental” life is entirely made up of ideas and attitudes towards
them. Imagination, memory, desire, thought, and belief all involve
ideas, and ideas are connected with suspended reactions. Ideas,
in fact, are parts of causes of actions, which become complete
causes when a suitable stimulus is applied. They are like explosives
waiting to be exploded. In fact, the similarity may be very close.
Trained soldiers, hearing the word “fire!” (which already existed
in them as an idea) proceed to cause explosions. The similarity of
language to explosives lies in the fact that a very small additional
stimulus can produce a tremendous effect. Consider the effects
which flowed from Hitler’s pronouncing the word “war!”

It is to be observed that words, when learnt, can become
substitutes for ideas. There is a condition called “thinking of”

SUSPENDED REACTIONS: KNOWLEDGE AND BELIEF

this or that, say water when you are in the desert. A dog appears,
from its behaviour, to be capable of being in this condition ; so
does an infant that cannot yet speak. When this condition exists,
it prompts behaviour having reference to water. When the word
“water” is known, the condition may consist (mainly, not wholly)
in the presence of this word, either overtly pronounced or merely
imagined. The word, when understood, has the same causal
efficacy as the idea. Familiar knowledge is apt to be purely verbal;
few schoolboys go beyond the words in reciting “William the
Conqueror 1066”. Words and ideas are, in fact, interchangeable;
both have meaning, and both have the same kind of causal
relations to what they mean. The difference is that, in the case of
words, the relation to what is meant is in the nature of a social
convention, and is learnt by hearing speech, whereas in the case
of ideas the relation is “natural”, i.e. it does not depend upon the
behaviour of other people, but upon intrinsic similarity and (one
must suppose) upon physiological processes existing in all human
beings, and to a lesser extent in the higher animals.

“Knowledge”, which is, in most forms, connected with sus-
pended reactions, is not a precise conception. Many of the
difficulties of philosophers have arisen from regarding it as
precise. Let us consider various ways of “knowing” the same fact.
Suppose that, at 4 p.m. yesterday, I heard the noise of an explosion.
When I heard it, I “knew” the noise in a certain sense, though
not in the sense in which the word is usually employed. This
sense, in spite of being unusual, cannot be discarded, since it is
essential in explaining what is meant by “empirical verification”.
Immediately afterwards, I may say “that was loud!” or “what
was that noise?” This is “immediate memory”, which differs
only in degree from sensation, since the physiological disturbance
caused by the noise has not yet wholly subsided. Immediately
before the explosion, if I have seen the train fired which leads to
a charge of explosive, I may be in a state of tense expectation;
this is, in a sense, akin to immediate memory, but directed to the
near future. Next comes true memory : I now remember the bang
I heard yesterday. My state is now made up of ideas (or images)
or words, together with belief and a context which dates the
occurrence remembered. I can imagine a bang just like , the one
that I remember, but when I do this, belief and dating are absent.
(The word “belief” is one which I shall discuss later.) Imagined

HUMAN knowledge: its scope and limits

events are not included in knowledge or error, because of the
absence of belief.

Sensation, immediate expectation, immediate memory, and true
memory all give knowledge which is, in some degree and with
appropriate limitations, independent of extraneous evidence. But
most of the knowledge of people with any degree of education is
not of any of these kinds. We know what we have been told or
have read in books or newspapers; here words come first, and it
is often unnecessary to realize what the words mean. When I
believe “William the Conqueror 1066”, what I am really believing
(as a rule) is: “the words ‘William the Conqueror 1066* are true”.
This has the advantage that the words can be made sensible
whenever I choose; the Conqueror is dead, but his name comes
to life whenever I pronounce it. It has also the advantage that the
name is public and the same for all, whereas the image (if any)
employed in thinking of William will differ from person to person,
and is sure to be too concrete. If (e.g.) we think of him on horse-
back, that will not suit “William was born at Falaise”, because
he was not born on horseback.

Sentences heard in narrative are, of course, not necessarily
understood in this purely verbal manner; indeed a purely verbal
understanding is essentially incomplete. A child reading an
exciting adventure story will “live through 5 ’ the adventures of the
hero, particularly if the hero is of about the same age as the reader.
If the hero leaps a chasm, the child’s muscles will grow taut; if
the hero sees a lion about to spring, the child will hold his breath.
Whatever happens to the hero, the child’s physiological condition
is a reproduction, on a smaller scale, of the physiological condition
of the hero. In adult life, the same result can be produced by good
writing. When Shakespeare’s Antony says “I am dying, Egypt,
dying”, we experience something which we do not experience
when we see in The Times a notice of the death of some person
unknown to us. One difference between poetry and bald statement
is that poetry seeks to take the reader behind the words to what
they signify.

The process called “verification” does not absolutely necessitate
(but often involves) an imaginative understanding of words, but
only a comparison of words used in advance with words used
when the fact concerned becomes sensible. You say “this litmus
paper will turn red”; I, later, say “this litmus paper has turned

112

SUSPENDED REACTIONS: KNOWLEDGE AND BELIEF

red”. Thus I need only pass outside the purely verbal region when
I use a sentence to express a present sensible fact.

“Knowledge” is a vague concept for two reasons. First, because
the meaning of a word is always more or less vague except in logic
and pure mathematics ; and second, because all that we count as
knowledge is in a greater or less degree uncertain, and there is
no way of deciding how much uncertainty makes a belief un-
worthy to be called “knowledge”, any more than how much loss
of hair makes a man bald.

“Knowledge” is sometimes defined as “true belief”, but this
definition is too wide. If you look at a clock which you believe
to be going, but which in fact has stopped, and you happen to
look at it at a moment when it is right, you will acquire a true
belief as to the time of day, but you cannot be correctly said to
have knowledge. The correct definition of “knowledge” need not
concern us at the moment ; what concerns us now is belief.

Let us take some simple sentence expressing something that is
or may be a sensible fact, such as “a loud bang is (or has been,
or will be) taking place”. We will suppose it a fact that such a
bang occurs at a place P at time t y and that the belief to be con-
sidered refers to this particular bang. That is to say we will amend
our sentence to “a loud bang occurs at place P at time t y \ We
will call this sentence S. What sort of thing is happening to me
when I believe this sentence, or rather when I believe what it
expresses ?

There are a number of possibilities. First, I may be at or near
the place P at the time t , and may hear the bang. In that case, at
time t I have direct sensible knowledge of it; ordinary language
would hardly call this “belief”, but for our purposes it is better
to include it in the scope of the word. Obviously this sort of
knowledge does not require words. No more does the immediate
memory that subsists while I am still shaken by the noise. But
how about more remote memory? Here, also, we may have no
words, but an auditory image accompanied by a feeling which
could be (but need not be) expressed in the words “that occurred”.
Immediate expectation also does not need words. When you
watch a door about to be slammed by the wind, your body and
mind are in a state of expectation of noise, and if no noise Resulted
you would experience a shock of surprise. This immediate
expectation is different from our ordinary expectations about

”3

HUMAN knowledge: its scope and limits

events that are not imminent. I expect that I shall get up to-
morrow morning, but my body is not in that unpleasant condition
in which it will be to-morrow morning when I am expecting to
get up in a moment. I doubt whether it is possible, without words,
to expect any event not in the immediate future. This is one of
the differences between expectation and memory.

Belief about something outside my own experience seems
usually only possible through the help of language, or some
rudimentary beginning of language. Sea-gulls and cannibals have
a “food-cry”, which in the cannibals is meant to give information,
but in the sea-gulls may be a spontaneous expression of emotion,
like a groan when the dentist hurts you. A noise of this sort is a
word to the hearer, but not to the utterer. An animars behaviour
may be affected by signs which have no analogy with language,
for instance when it is in search of water in an unknown region.
If a thirsty animal runs persistently down into a valley, I should
be inclined to say that it “believes” there is water there, and in
such a case there would be non-verbal belief in something that is
as yet outside the animal’s experience. However, I do not wish
to become involved in a controversy as to the meaning of words,
so I will not insist upon the view that such behaviour shows
“belief”.

Among human beings, the usual way of acquiring beliefs as to
what has not been, and is not just about to be, experienced is
through verbal testimony. To revert to our sentence S, some
person whom we believe to be truthful pronounces it in our
presence, and we then believe what the sentence asserts. I want
to inquire what is actually occurring in us while we are believing
the sentence.

We must, of course, distinguish a belief as a habit from the
same belief when it is active. This distinction is necessary in
regard to all habits. An acquired habit consists in the fact that a
certain stimulus, whenever it occurs, now produces a certain
reaction which it did not produce in the animal in question until
the animal had had certain experiences. We must suppose that,
even in the absence of the stimulus concerned, there is some
difference between an animal that has a certain habit and one
that lacks it. A man who understands the word “fire” must differ
in some way from a man who does not, even when he is not hearing
the word. We suppose the difference to be in the brain, but its

114

SUSPENDED REACTIONS! KNOWLEDGE AND BELIEF

nature is hypothetical. However, it is not a habit as a permanent
character of an organism that concerns us, but the active habit,
which is only displayed when the appropriate stimulus is applied.
In the case we are investigating, the stimulus is the sentence S;
or rather, since the sentence may have been never heard before,
and may therefore have had no chance to generate a habit, the
stimulus is the succession of the words composing S, each of
which, we are supposing, is familiar to the hearer, and has already
generated the habit which constitutes understanding of its
meaning.

It may happen, when we hear a sentence, that we do not trouble
to think what it means, but merely believe “this sentence is true”.
With certain kinds of sentences, this is the usual reaction; for
example, when we are told someone’s address and we only wish
to write to him. If we wish to go and see him, the meaning of the
words becomes important, but for sending him a letter the words
alone are sufficient. When we believe “this sentence is true”, we
are not believing what the sentence asserts; if the sentence is in
a language unknown to us, we may believe that it is true without
being able to find out what it asserts — for example, if it is a
sentence in a Greek Testament and we know no Greek. I shall
therefore ignore this case, and consider what happens when,
hearing S, we believe what S asserts.

Let us somewhat simplify the sentence, and suppose that, when
I am walking with a friend, he says: “There was an explosion
here yesterday.” I may believe him, or understand him without
believing. Let us suppose that I believe him, and that I believe
what his words assert, not merely that the words are true. The
most important word in the sentence is “explosion”. This word,
when I am actively understanding it, rouses in me faint imitations
of the effects of hearing an actual explosion — auditory images,
images of nervous shock, etc. Owing to the word “here”, these
images are combined in my mental picture with the surrounding
scenery. Owing to the word “yesterday”, they are combined with
recollections of yesterday’s experiences. All this, so far, is involved
in understanding the sentence, whether or not it is believed. I
incline to the view that believing a sentence is a simpler occurrence
than understanding without belief ; I think the primitive .reaction
is belief, and that understanding without belief involves inhibition
of the impulse to belief. What distinguishes belief is readiness foi

n 5

human knowledge: its scope and limits

any action that may be called for if what is asserted is a fact.
Suppose for instance that an acquaintance of mine has dis-
appeared, and is known to have been hereabouts yesterday, then
belief may prompt me to search for signs of his remains, which I
shall not do if I understand without believing. If no such action
is called for, there is at least the action of repeating what I have
been told whenever it may seem appropriate to do so.

From all this it appears that, when I believe what a certain
sentence asserts, the words, having had their intended effect,
need no longer be present to me. All that need exist is a state of
mind and body appropriate to the fact that the sentence asserts.

It is an error to suppose that beliefs consist solely in tendencies
to actions of certain kinds. Let us take an analogy; a belief may
be compared to a cistern plus a pipe plus a tap. The tap can be
turned on, and the belief can influence action, but neither happens
without an additional stimulus. When a man is believing some-
thing, there must exist in him cither appropriate words or
appropriate images, or, at the very least, appropriate muscular
adjustments. Any of these, given certain additional circumstances
(which correspond to turning on the tap), will produce action,
and this action may be such as to show an outside observer what
it is that is being believed; this is particularly the case if the
action consists in pronouncing appropriate words. The impulse
to action, given the right stimulus, is inherent in the presence of
words, images, or muscular adjustments. To entertain an idea
vividly and not act upon it is difficult. If, alone at night, you read
a story in which a man is stabbed in the back, you will have an
impulse to press your chair tight against the wall. Booth the actor
(the brother of Lincoln’s assassin), on one occasion when he was
playing Macbeth under the influence of liquor, refused to be
killed, and chased Macduff murderously all through the stalls.
It is unwise to read a ghost story just before walking through a
churchyard at midnight. As these examples show, when an idea
is entertained without belief, the impulse to belief is not absent,
but is inhibited. Belief is not something added to an idea previ-
ously merely entertained, but something subtracted from an
idea, by an effort, when the idea is considered without being
accepted.

Another example is the difficulty that uneducated people feel
about hypotheses. If you say “let us suppose so-and-so and see

116

SUSPENDED REACTIONS! KNOWLEDGE AND BELIEF

what comes of the supposition”, they will tend either to believe
what you suppose, or to think that you are wasting your time.
For this reason, reductio ad absurdum is a form of argument that
is repugnant to those who are not familiar with logic or mathe-
matics ; if the hypothesis is going to be proved false, they cannot
make themselves hypothetically entertain it.

I do not wish to exaggerate the scope of pre-linguistic belief:
only very simple and primitive matters can be dealt with in the
absence of words. Words are public, permanent (when written),
and capable of being created at will. These merits make it possible
to have more complicated habits based on words than any that
could be based on wordless ideas or images. By acquiring verbal
habits we can prepare ourselves for actual situations when they
arise. What is more, knowledge can be externalized in books of
reference, and need then only exist in human beings when it is
wanted. Consider the telephone book: no one wants to know all
its contents, or indeed any except at certain moments. The people
who compile the book may never use it, and the immense majority
of those who use it have had no hand in compiling it. This kind
of socialized potential knowledge is only rendered possible by
language, in fact by written language. All that the user of the
telephone needs to know is a simple prescription for deriving
appropriate action from the appropriate entry in the book. By
such devices we diminish enormously the amount of knowledge
that it is necessary to carry in our heads.

All generalized knowledge is of this sort. Suppose the geography
book tells me that Semipalatinsk is a province and city of Central
Asia, in the territory of the U.S.S.R. This knowledge will remain
purely verbal unless I have occasion to go to Semipalatinsk, but
if this should happen there are rules by which the words of the
book show me how to produce desired experiences. In such a
case I may be said to understand the words if I know what action
they prescribe when I have desires connected with what the
words mean, or, in an extreme case, merely a desire to know what
the words mean. You may feel a longing to see the Altai Mountains,
knowing nothing about them except that that is their name. In
that case, the guide book shows you what you must do in order
to know the proposition: “ These are the Altai Mountains. v When
you have learnt arithmetic you can deal with all the innumerable
occasions on which you have to count your change in shops, but

n 7

HUMAN knowledge: its scope and limits

in learning arithmetic you need not be thinking of its applications.
In 9uch ways the province of purely verbal knowledge becomes
wider and wider, and at last it becomes easy to forget that verbal
knowledge must have some relation to sensible experience. But
except through such relation we cannot define empirical truth and
falsehood, and to forget it is therefore fatal to any hope of a sound
philosophy.

118

Chapter VI
SENTENCES

I want in this chapter to consider sentences as opposed to
words, and to ask in what consists the understanding of words
that do not denote objects, but occur only as parts of sentences.
We saw that the one word “water” may be used to express what,
if fully expressed, requires different sentences. It may mean
“here is water”; it may mean “I want water”; it may, if pro-
nounced with an interrogative inflexion, mean “is this water?”
Obviously such ambiguity is not desirable, especially in writing,
where differences of inflexion are difficult to indicate. We there-
fore need such words as “here is”, “I want”, “is this”. It is the
function of such words that forms the theme of this chapter.

Consider the following sentences: “there is fire here”, “there
was fire here”, “there will be fire here”, “is there fire here?”
“I want fire here”, “there is no fire here”. These sentences are
respectively present, past, future, interrogative, optative, and
negative, but all deal with the same object, namely fire.

The word “fire” may be caused in me in various ways. When it
is caused by the sensible presence of fire, I communicate the fact
by the sentence “there is fire here”; when by the memory, by
the sentence “there was fire here”. But I may use this sentence,
not to express a memory, but to report what I have been told,
or to state an inference from charred embers. In the former case,
the word “fire” is caused in me by my hearing the word; in the
latter case, by my seeing something which I know to be an effect
of fire. Thus when I say “there was fire here”, my state of mind
may be one of several very different possibilities. In spite of these
subjective differences, however, what I am asserting is the same
in all the different cases. If my assertion is true, a certain occurrence
took place here, and the occurrence in virtue of which it is true
is the same whether the occurrence is remembered or known
through testimony or inferred from present traces of past com-
bustion. It is for this reason that we use the same words in these
various cases, for a sentence in the indicative is concerned, not to
express a state of mind (though it always does so), but to assert
a fact other than that expressed by the sentence. But we will

119

HUMAN knowledge: its scope and limits

postpone the explicit consideration of truth and falsehood to a
later chapter.

Similar subjective ambiguities exist in connection with the
sentence “there will be fire here”. In a situation in which you
experience immediate expectation of fire, your subjective state is
analogous to memory, except in the vital point of being directed
to the future. But as a rule statements about the future are
inferences. You may see a damp haystack fermenting and infer
that it will burn, or you may have been told that at some future
date there is to be a bonfire here. But again these various possi-
bilities make no difference to what is asserted when you say “there
will be fire here”.

“Is there fire here ?” may be a form of imperative, or a suggestion
for investigation. This sentence does not make an assertion, but
shows a desire to be able to make one. The difference from “there
is a fire here” is not in anything having an external reference, but
in our attitude to what has such a reference. We may say that
there is an “idea” called “fire-hcre-now” ; when we preface these
words by “there is” we assert this idea, whereas when we preface
them by “is there” we “actively consider” them, i.e. we are
concerned to find out whether or not to assert them. I speak of
“outward reference” in a preliminary way, as the concept is a
difficult one, demanding considerable discussion.

“I want fire here” is a sentence in the indicative, asserting that
I feel a certain desire, but it is commonly used as if it were an
expression of desire, not an assertion of it. Strictly speaking, desire
ought to be expressed by “would there were a fire here” or “Oh
for a fire here!” This is more easily and naturally expressed in a
language which, like Greek, has an optative mood. The sentence
“Oh for a fire!” asserts nothing, and is therefore neither true nor
false. It expresses a desire, and a person hearing me pronounce it
may infer that I feel a desire, but it does not assert that I feel a
desire. Similarly when I say “there is a fire here” I express a
belief, and the hearer can infer that I have this belief, but I do
not assert that I have a belief.

When I say “there is not a fire here”, what may be called the
“content” is the same as w'hen I say “there is fire here”, but this
content is denied instead of being asserted.

Reviewing the above sentences, but omitting those referring to
past or future, we find that, considering what they express, they

120

SENTENCES

all have the same core, namely “fire-here-now”. The ideas
expressed by “fire”, “here”, and “now” may be called “indica-
tive” ideas, that is to say, they can all indicate features of a
sensible experience. Taken all together, they constitute one
complex indicative idea. An indicative idea sometimes indicates,
and sometimes does not; if there is a fire here now, “fire-here-
now” indicates that fire, but if there is not a fire, “fire-here-now”
indicates nothing. Towards an indicative idea we may have
various attitudes: assertive, interrogative, optative, or negative.
These attitudes are expressed by the words “there is”, “is there”,
“oh for”, and “there is not” respectively. (I do not pretend that
this list of possible attitudes is exhaustive.) These attitudes, which
are expressed by the above words, can also be asserted , but we then
need indicative words for them; the words are “belief”, “doubt”,
“desire”, “disbelief”. This leads to new sentences, all of which
are assertions, but about my state of mind, not about fire. The
sentences are: “I believe there is a fire here now”, “I wonder if
there is a fire here now”, “I hope there is a fire here now”, and
“I disbelieve that there is a fire here now”.

It is evident that “thcre-is”, “is-thcrc”, “oh-for”, and “there-
is-not” should each be regarded as one word, and as expressing
different attitudes on the part of the speaker to one and the same
idea. It is not their function to indicate objects, as names do; the
fact that the word “not” can be used significantly does not imply
that there is an object called “not” in some Platonic heaven. For
the understanding of language it is essential to realize that, while
some necessary words mean objects, others do not.

Words that mean objects may be called “indicative” words. I
include among such words not only names, but words denoting
qualities such as “white”, “hard”, “warm”, and words denoting
perceptible relations such as “before”, “above”, “in”. If the sole
purpose of language were to describe sensible facts, we could
content ourselves with indicative words. But, as we have seen,
such words do not suffice to express doubt, desire, or disbelief.
They also do not suffice to express logical connections, e.g. “if
that is so, I’ll eat my hat”, or “if Wilson had been more tactful,
America would have joined the League of Nations”. Nor do they
suffice for sentences needing such words as “all” and *' ‘some”,
“th/e” and “a”. The significance of words of this kind can only be
explained by explaining the significance of sentences in which

121

HUMAN knowledge: its scope and limits

they occur. When you want to explain the word “lion,” you can
take your child to the Zoo and say “look, that’s a lion.” But there
is no Zoo where you can show him if or the or nevertheless , for
these are not indicative words. They are needed in sentences, but
only in sentences not concerned exclusively with the assertion of
single facts. It is because we need such sentences that words
which are not indicative are indispensable.

122

Chapter VII

EXTERNAL REFERENCE OF IDEAS
AND BELIEFS

T he kind of external reference with which we shall be
concerned in this chapter is not that by which experiences
are interpreted as percepts of external objects, as when,
for instance, a visual sensation produces in me a condition called
“seeing a table”. This kind of external reference will be considered
in connection with the interpretation of physics and the evidence
for its truth. What we are now concerned with is a reference of
one part of my mental life to another, and only derivatively to
things not forming part of my experience.

We are in the habit of saying that we think of so-and-so and
that we believe in such-and-such. It is the meaning of “of” and
“in” in such phrases that I wish to discuss, as a necessary pre-
liminary to the definition of “truth” and “falsehood”.

We considered in an earlier chapter the process of ostensive
definition as the source of the meanings of words. But we then
found that a given word can “mean” an idea as well as a sensible
experience; this happens, notably, when the word is used to
express a memory. When the same word can be used to denote
an idea or a sensible experience, that is a sign that the idea is an
idea “of” the sensible experience. But obviously the relation
expressed by this word “of” is one which can exist independently
of language, and is, in fact, presupposed in the use of the same
word for an idea and a sensible experience.

The relation with which we are concerned is perhaps seen most
clearly in the case of memory. Suppose you have lately seen
something horrible — say a friend run over and killed by a lorry.
You will have a constantly recurring picture of the event in your
mind, not only as pure imagination, but as something that you
know actually occurred. As the dreadful swift sequence over-
whelms you once more, you may say to yourself “yes, that really
happened”. But in what sense can this be true? For your recol-
lections are now , and consist of images, not of sensations, still less
of actual motor lorries. The sense in which it is nevertheless true
is w r hat we have to elucidate.

123

HUMAN knowledge: its scope and limits

Images occur in two ways, as imagination and as recollection.
I have sometimes, under the influence of fatigue or fever, seen
the faces of people of whom I was fond, not with the benign
expression to which I was accustomed, but horribly grimacing
and grotesque. These painful images did not command belief
unless my temperature was high enough to cause delirium. Even
in deliberate recollection there are often imaginative accretions
which are not believed, but these do not count as memories.
Whatever counts as a memory consists of images or words which
are felt as referring to some earlier experience. Since it is clear
that words can only express memories because a given word can
apply both to an image (or idea) and to a sensible occurrence, it
is clear that we must first consider non-verbal memory, with a
view to discovering what is the relation of an idea to a sensible
experience which leads us to use the same word for both. I shall
therefore, for the present, exclude memories expressed in words,
and consider only those that come as images accompanied by the
belief or feeling that they refer to a previous occurrence.

Suppose I am asked to describe the furniture of my room. I
may go to my room and record what I see, or I may call up a
picture of my room and record what I see with my mind's eye.
If I am a good visualizer and my room is one which I have
inhabited for some time, the two methods will give results that,
at least in broad outline, will be indistinguishable. It is easy, in
this way, to test the accuracy of my memory. But before it is
tested I implicitly believe it. Some memories are not capable of
being tested at all thoroughly, for instance if you have been the
sole spectator of a murder; nevertheless your evidence will be
accepted unless there is reason to suspect you of perjury. At
present, it is not the trustworthiness of memory that concerns
us, but the analysis of the occurrence.

What is involved in saying that A is an “image" or “idea" of
B ? First, there must be resemblance ; more particularly, if both
are complex, there must be resemblance of structure. Second, B
must play a certain definite part in the causation of A. Third,
A and B must have certain effects in common, for example, they
can cause the same words to occur to a person who experiences
them. When these three relations exist, I shall say that B is the
“prototype" of A.

But if A is a recollection of B, something more is involved.

124

EXTERNAL REFERENCE OF IDEAS AND BELIEFS

For in this case A is felt or believed to be pointing to something
other than itself, and this something is, in fact, B. We should like
to say that A is felt to be pointing to B, but this we have no right
to say, since B is not itself present to the person recollecting;
what is present is only A, as B’s representative. We must say,
therefore, that, in memory as opposed to pure imagination, there
is the belief: “A is related to something as idea to prototype”,
where the relation of idea to prototype is defined by the three
characteristics mentioned in the preceding paragraph. I do not
mean, of course, that an ordinary memory belief has the explicit-
ness suggested by the above analysis. I mean only that, in memory,
an idea is vaguely felt to point beyond itself, and that the above
is an account of what may be the actual state of affairs when this
vague feeling is justifiable.

When B is the prototype of A, we say that A is an image “of”
B. This is a definition of this use of the word “of”.

It is obvious that A may be an image of B without the person
concerned being aware of the fact. It is also obvious that A may
have many prototypes. If I tell you I met a negro in an English
country lane, the word “negro” may call up in your mind an
image vaguely compounded of many negroes whom you have
seen; in this case they must all count as prototypes of your image.
In general, even when an image has only one prototype, it will
usually be vaguer than its prototype. If, for example, you have
an image of a shade of colour, various shades that you can dis-
tinguish when sensibly present might all serve as its prototypes.
This, incidentally, supplies an answer to Hume’s query: Could
you imagine a shade of colour you had never seen, if it was inter-
mediate between two very similar shades that you had seen?
The answer is that you could not form so precise an image, even
of a colour that you had seen, but that you could form a vague
image, equally appropriate to the shade that you had not seen
and to the two similar shades that you had seen.

It will be seen that, according to the above theory, the external
reference of an idea or image consists in a belief, which, when
made explicit, may be expressed in the words: “this has a proto-
type”. In the absence of such a belief (which, when it exists, is
usually a somewhat vague feeling), although there may be 'in fact
a prototype there is no external reference. This is the case of pure
imagination.

125

HUMAN knowledge: its scope and limits

In the case of a memory- belief, if what is said to be remembered
is an experience of the person remembering, the above kind of
external reference is the only kind required. But as a rule there
is also another kind, namely that which, at the beginning of this
chapter, we declined to consider. Suppose I remember “I saw
an elephant yesterday”. There is involved not only my experience
of yesterday, but belief in an animal having an independent
existence, not only when I saw it, but also before and after. All
this depends upon animal inference in the sense to be considered
in Part III, involving a reference external, not only to my present
experience, but to the whole of my experience. This kind of
external reference, however, takes us beyond the subject of the
present chapter.

126

Chapter VIII

TRUTH: ELEMENTARY FORMS

T ruth and falsehood, in so far as they are public, are
attributes of sentences, either in the indicative or in the
subjunctive or conditional. In the present chapter, which
will consider only the simpler examples of truth, I shall confine
myself to sentences in the indicative. In addition to sentences
there are some other ways of making public statements, maps, for
instance, and graphs. There are also conventional devices for
reducing a sentence to one essential word, as is done in telephone
books and railway time tables. But for our purposes we may,
without any important loss of generality, confine ourselves to
fully expressed sentences. And until we have considered logical
words, which will be the subject of the next two chapters, we
must confine ourselves to sentences in the indicative.

But in order to define “truth” and “falsehood” we must go
behind sentences to what they “express” and what they “indicate”.

A sentence has, to begin with, a properly which I shall call
“signification”. This is the property which is preserved in an
accurate translation. “Two and two make four” has the same
signification as “deux et deux font quatre”. Signification is also
preserved when the wording is changed; c.g. “A is the husband
of B”, “B is the wife of A”, “A is a male who is married to B”,
“A is married to B, who is a female”, all have the same significa-
tion. It is obvious that when two sentences have the same significa-
tion both are true or both are false; therefore whatever dis-
tinguishes truth from falsehood is to be sought rather in the
signification of sentences than in sentences themselves.

Some sentences which, at first sight, appear to be correctly
constructed, are in fact nonsense, in the sense that they have no
signification. Such, if interpreted literally, are “necessity is the
mother of invention” and “procrastination is the thief of time”.
A very important part of logical syntax consists of rules for
avoiding nonsense in constructing sentences. But for the present
we are concerned with sentences that are too simple to run the
risk of being nonsensical.

To arrive at what a sentence “signifies”, the easiest way is to

127

HUMAN knowledge: its scope and limits

ask ourselves what is in common between a sentence in one
language and its translation into another. Suppose that on a
given occasion I say to an Englishman “I am hot” and to a
Frenchman “j’ai chaud”, the two sentences express the same
state of mind and body, and are made true (or false) by the same
fact. The signification of a sentence would thus seem to have
two aspects: on the one hand it “expresses” the condition of the
person uttering it, and on the other hand it points outside this
present condition to something in virtue of which it is true or
false. What an asserted sentence expresses is a belief ; what makes
it true or false is a fact , which is in general distinct from the belief.
Truth and falsehood are external relations, that is to say, no
analysis of a sentence or a belief will show whether it is true or
false. (This does not apply to logic and mathematics, where truth
or falsehood, as the case may be, follows from the form of the
sentence. But I am for the present ignoring logical truth.) Con-
sider e.g. the sentence “I am an uncle”, and suppose you know
that your sister in India is due to have a child, but you do not
know whether the child has yet been born. No analysis of the
sentence or of your state of mind will show whether the sentence
is true or false, since its truth or falsehood depends upon events
in India as to which you are in ignorance. But although under-
standing the sentence does not enable you to know whether it is
true or false, it does enable you to know what sort of fact would
make it true and what sort would make it false; this, therefore, is
part of the signification of a sentence, or is at least inseparably
connected with the signification, although the actual truth or
falsehood (as the case may be) is not.

If “truth” and “falsehood” had been defined, we could say
that two sentences are to have, by definition, the same “significa-
tion” if whatever possible state of affairs makes one of them true
also makes the other true, and vice versa. But, as we shall see, it
is not clear that “truth” and “falsehood” can be defined without
first defining “signification”.

There are, we said, two sides to signification, which we may
call subjective and objective respectively. The subjective side has
to do with the state of the person uttering the sentence, while the
objective side has to do with what would make the sentence true
or false. Let us begin by considering the subjective side.

When we say that a sentence is true, we mean that a person

128

truth: elementary forms

asserting it will be speaking truly. A person may pronounce a
sentence without intending to assert it: when an actor says “this
is I, Hamlet the Dane”, no one believes him, but no one accuses
him of lying. This shows that the subjective side in the analysis
of signification is essential. When we say that a sentence is “true”,
we mean to say something about the state of mind of a person
uttering or hearing it with belief. It is in fact primarily beliefs
that are true or false ; sentences only become so through the fact
that they can express beliefs. It is therefore in beliefs that the
subjective side of the signification of sentences is to be sought.

We may say that two sentences have the same signification if
they express the same belief. But we must, having said this,
explain in what sense two people (or one person at different
times) can have the same belief, and by what tests we can dis-
cover when this is the case. For practical purposes we may say
that two people who speak the same language have the same
belief if they accept the same sentence as expressing it ; and when
two people speak different languages, their beliefs are the same
if a competent interpreter regards the sentence in which one of
them expresses his belief as a translation of that used by the other.
But this criterion is not theoretically sufficient, since infants that
cannot speak must be allowed to have beliefs, and so (I should
say) must animals.

“Belief”, as I wish to use the word, denotes a state of mind
or body or both, in which an animal acts with reference to some-
thing not sensibly present. When I go to the station in expectation
of finding a train, my action expresses a belief. So does the action
of a dog excited by the smell of fox. So does that of a bird in a
room, which flies against the window panes in the hope of getting
out. Among human beings, the only action by which a belief
is expressed is, very often, the pronouncing of appropriate words.

It will be seen that, according to the above definition of “belief”,
it is closely connected with meaning and with ostensive definition.
Words have “meaning” when there is an association or a con-
ditioned reflex connecting them with something other than them-
selves — this, at least, applies to indicative words. I say “look,
there’s a fox”, and you act as you would do if you smelt a fox.
I say “fox” when I see a fox, because a fox suggests the word
“fox” as well as vice versa. When, the fox having just disappeared,
I utter the word “fox”, and when you, having not yet seen the

129 1

human knowledge: its scope and limits

animal, hear the word, there is “belief” in the sense defined above.
So there is when, without speaking, you look for the fox. But it
is only when action is suspended that belief becomes a definite
state of mind — for instance when you have just looked up a train
that you mean to take to-morrow. When immediate action is called
for, energy may be drained into the muscles, and “belief” may
be shown as merely a characteristic of bodily movements. But it
must be remembered that shouting “fox” or “tally-ho” is a bodily
movement; we cannot therefore deny that bodily movements may
express beliefs.

External reference, which we discussed in the last chapter,
exists in all indicative words when used in the way in which the
use of words begins. It exists also in non-verbal behaviour, as
when a dog scratches at a rabbit- hole because he has seen a rabbit
go down the hole. But when behaviour is non-verbal it is difficult
for the observer, and often for the agent, to say what, exactly, it
is that the behaviour refers to. Words, like balances and thermo-
meters, are instruments of precision, though often not very good
ones; but that to which they give precision can exist, and be
apprehended vaguely, without their help.

To put the matter schematically, with a more or less unreal
simplification : the presence of a stimulus A causes a certain kind
of behaviour, say B; as a result of experience, something else,
say C, may cause B in the absence of A. In that case, C may be
said to cause “belief” in A, and “belief” in A may be said to be
a feature of the behaviour B. When words come in, all this
becomes more precise. The sight of a fox (A) causes you to
pronounce the word “fox” (B); you may learn the trail of a fox
in snow (C), and, seeing it, say “fox”. You are then “believing”
A because of C. And if the trail was made by a fox, your belief
is true.

That which has external reference — the belief or idea or bodily
movement — is in some cases public and in others private. It is
public when it consists in overt behaviour, including speech; it
is private when it consists of images or “thoughts”. (The meaning
of “public” and “private” in this connection will be explained
in Part III.) When an occurrence in an organism has external
reference, the only feature always present is the causal one
explained in the last paragraph, namely that the occurrence has
some of the effects that would result from the sensible presence

130

truth: elementary forms

of that which is its external reference. We will give the name
“representational occurrence” to anything that happens in an
organism and has external reference.

In addition to the essential causal relation by which “repre-
sentational occurrence” is defined, further relations exist in certain
kinds of such occurrences. In a memory image there is resemblance
to what is represented (i.e. remembered). In other images there
is likely also to be resemblance, though of a less exact kind. If
somebody tells you “your son has been killed by falling over a
precipice”, you are likely to have a very vivid image which will
be correct in some respects but not in others. But words (except
when onomatopoeic) have no resemblance to what they mean,
and therefore verbal beliefs cannot be judged true or false by
likeness to, or difference from, what they assert. Verbal behaviour
is only one form of bodily behaviour that is representational;
another form is that of the dog scratching at the rabbit-hole. We
may say quite generally that bodily behaviour, when represen-
tational, need not have any resemblance to that to which it refers.

Nevertheless, in the case of explicit language, there is often
a structural resemblance between a sentence and what it asserts.
Suppose you see a fox eat a goose, and afterwards you say “the
fox ate the goose”. The original occurrence was a relation between
a fox and a goose, while the sentence creates a relation between
the word “fox” and the word “goose”, namely that the word
“ate” comes between them. (Cf. Wittgenstein’s Tractatus.) This
possible structural similarity between a sentence and what it
asserts has a certain importance, but not, I think, an importance
which is fundamental.

The above account of what makes a representational occurrence
“true” is, I think, correct when it is applicable, but there are
various extensions which give “truth” a wider scope.

Let us begin with memory. You may recollect an event which
calls for no present action, and in that case the above definition
of “true” is not applicable. Your memory, if it is in images, may
then be “true” in the sense of being like the event. And even if
no present action is called for, there may be future situations in
which your memory has practical importance, and it may now
be called “true” if it will then fulfil the test.

But what is of more importance is what may be called “deriva-
tive” meaning, which is a property of sentences whose several

131

human knowledge: its scope and limits

words have “primary” meaning. Suppose that, for a given child,
the words “cat”, “dog”, and “hate” have primary meaning, in
the sense that they have been learnt by ostensive definition. Then
the sentence “cats hate dogs” has a meaning which does not have
to be learnt by a fresh process of either ostensive or nominal
definition. It is, moreover, a sentence which can never be verified
by one sensible occurrence; in this it differs from “there will be
a loud noise in a moment”. Only in Plato’s heaven could we see
THE CAT hating THE DOG. Here on earth, the facts in virtue
of which the sentence is true are many, and cannot all be expe-
rienced at one time. The relation of the sentence to the facts in
virtue of which it is true is derivative from a number of other
sentences, each of the form: “This is a cat and that is a dog,
and this hates that.” (I am taking “hate” as a characteristic of
overt behaviour; I am doing this not as a theory, but for purposes
of illustration.) We have here three sentences, (a) “this is a cat”,
(b) “that is a dog”, ( c ) “this hates that”. Each of these can be
directly caused by the present sensible facts, given that the
observer has learnt English. A sufficient number of such sen-
tences, or of the corresponding observations or beliefs, will, in
most people, in time cause the sentence “cats hate dogs”, of
which the meaning follows by the laws of syntax from the meanings
of sentences of the forms (a), (6), and (c). It is in this sense that
the meaning of such sentences is “derivative”. For the present,
having observed that the meaning of most sentences is derivative,
I wish to confine myself to sentences of which the meaning is
primary.

Let us consider the sentence “this is a cat”, uttered when a
cat is sensibly present. Hitherto I have been considering “truth”
and “falsehood” as ideas only applicable to representations of
things not sensibly present, and if we adhered strictly to this view
our sentences (a), ( b) y and (c), when uttered, would be neither true
nor false. This way of using the words would, however, be incon-
venient, and I should prefer to say that ( a ), (£), and (c) are true
or false.

If, in the presence of an animal, I say “this is a cat”, what are
the possibilities of falsehood? There is, first, deliberate lying: I
may be talking to a blind man, and wish him to think it a cat
when it is really a rabbit. We may exclude this case, on the
ground that the words I utter do not express a belief, and on the

332

truth: elementary forms

further ground that the word “cat” is not caused by what I see,
but by some ulterior motive. Then there is the case where I do
not see distinctly, owing to darkness or bad eyesight, and when
some one turns on another light I say “oh, I see it was not a
cat”. In this case what I see must have some likeness to a cat,
and if I had said “this is something resembling a cat” I should
have spoken truly. Then there is the case of insufficient know-
ledge of the language, leading me to give the name “cat” to what
is officially called a puma. In this case there is only social error:
my language is not correct English, but in the language that I
speak my statement is true. Finally, I may be suffering from
delirium tremens and see a cat where there is nothing, at least
from a public point of view. In the absence of such unusual
possibilities, my statement “this is a cat” will be true.

When there is a cat, and I say “there is a cat”, what is the
relation of what I say to the actual present quadruped ? There
is a causal relation: the sight of the cat causes the word “cat”,
but this, as we saw in the case of indistinct vision, is not enough
to insure truth, since something not a cat may cause the word
“cat”. When I say “this is a cat”, I am asserting the existence
of something which is not merely a momentary visual experience
of my own, but lives and breathes and mews and purrs and is
capable of feline joys and sorrows. All this is erroneous in the
case of delirium tremens. Let us therefore take a simpler example,
say “this is blue”. This statement need not imply anything beyond
an experience private to me, and is therefore not liable to the
kind of error that afflicts the drunkard. In this case, the only
possibility of error is ignorance of the language, leading me to
call “blue” what others would call “violet”. This is social error,
not intellectual error; what I am believing is true, but my words
are ill chosen. In this case, therefore, the possibility of genuine
falsehood in my statement seems to be at a minimum.

We may say generally: an indicative word is true when it is
caused by what it means, assuming that the word is used in what
may be called the exclamatory manner, as when people shout
“fire I” or “murder!” In developed speech, we usually drop this
way of using indicative words, and, instead, preface the word
by “this is”. Thus the statement “this is blue” is true if it is
caused by what “blue” means. This is in fact a tautology. But
most words, such as “cat” and “dog”, mean not only what can

133

HUMAN knowledge: its scope and limits

be a momentary percept, but also the habitual concomitants of
this kind of percept. If these are only usual but not invariable
concomitants, there may be error in using the word that the
percept causes; this is the case of the victim of delirium tremens ,
and also of Isaac when he mistook Jacob for Esau. Most words
embody animal inductions which are usually true, but not always;
this applies, more particularly, to names of objects or kinds of
objects, such as our friends or the various species of animals.
Whenever such words are employed as a result of a percept there
is, therefore, some possibility of error, though often only a very
slight one.

We may now say, as a definition: A sentence of the form “this
is A” is called “true” when it is caused by what “A” means. We
may say further that a sentence of the form “that was A” or
“there will be A” is “true” if “this is A” was, or in the second
case will be, true in the above sense. This covers all sentences
asserting what are, were, or will be, facts of perception, and also
those in which, from a percept, we correctly infer its usual con-
comitants by animal inference, in so far, at least, as such con-
comitants can form part of the meaning of an indicative word.
This covers all the factual premisses of empirical knowledge. It
does not cover general statements, such as “dogs bark”, nor yet
principles of inference, whether deductive or non- demonstrative.
These cannot be adequately considered until we have dealt with
the meaning of logical words such as “or” and “all”. The above,
moreover, is only a definition of “truth”, not of “falsehood”.
“Falsehood” remains to be dealt with later.

There is one important observation to be made about our
definitions of “meaning” and “truth”, and that is that both depend
upon an interpretation of “cause” which, according to modern
physics, might seem to be crude and only partially applicable
to natural processes. If this view is adopted, it follows that what-
ever defects belong to this old-fashioned notion of “cause” belong
also to the notions of “meaning” and “truth” as we have inter-
preted them. I do not think, however, that this is a very serious
objection. Both concepts, on other grounds, are necessarily some-
what vague and inexact, and these other grounds do much more
to prevent precision than is done by modem physics. Such
propositions as “lightning causes thunder”, “micro-organisms
cause fevers”, “wounds cause pain”, although they have not the

*34

truth: elementary forms

certainty formerly attributed to them, and even if (what for
reasons that will appear later, I do not believe) “cause” is a
rough-and-ready notion belonging to a certain stage of science,
not a fundamental category as used to be thought, nevertheless
express in a convenient form truths about the usual though not
invariable course of nature, and as such are still useful except
where, as in quantum physics, the last refinement of accuracy
is sought in spite of its complication and its consequent useless-
ness for most purposes of prediction. If human behaviour could
be calculated by the physicists, we should have no need of such
concepts as “meaning”, “belief”, and “truth”. But in the mean-
time they remain useful, and up to a point they can be freed
from ambiguity and vagueness. Beyond this point it would be
useless to attempt to go, if “cause” is in fact not a fundamental
concept of science. But if, as I hold, the concept of “cause”
is indispensable, then the above considerations do not arise, or at
any rate arise only in a modified form.

*35

Chapter IX

LOGICAL WORDS AND FALSEHOOD

I N the preceding Chapter we dealt with the truth of beliefs
and sentences in cases where this depends only upon ob-
servation and not upon inference from previous knowledge.
In this Chapter we have to begin the inquiry into sentences
of kinds that can be proved or disproved when suitable data
derived from observation are known. Where such sentences
are concerned, we no longer have to consider the relation of
beliefs or sentences to something which is in general neither a
belief nor a sentence; we have instead to consider only syntactical
relations between sentences, in virtue of which the indubitable
or probable truth or falsehood of a certain sentence follows from
the truth or falsehood of certain others.

In such inferences there are certain words, which I shall call
“logical” words, of which one or more always occur. These words
are of two kinds, which may be called respectively “conjunctions”
and “general words”, though not quite in the usual grammatical
sense. Examples of conjunctions are “not”, “or”, “and”, “if-
then”. Examples of general words are “all” and “some”. (Why
“some” is called a “general” word will appear as we proceed.)

By the use of conjunctions we can make various simple in-
ferences. If “p” is true, “not-p” is false; if “p” is false, “not-p”
is true. If “p” is true, “p or q y is true; if “<7” is true, “p or <7”
is true. If “p” is true and “<7” is true, “p and q” is true. And so
on. Sentences containing conjunctions I shall call “molecular”
sentences, the “p” and “<7” which are conjoined being conceived
as the “atoms”. Given the truth or falsehood of a set of proposi-
tions, the truth or falsehood of every molecular proposition con-
structed out of the set follows by syntactical rules, and requires
no fresh observation of facts. We are, in fact, in the domain of
logic.

Given that we know about “p”, both what is involved in
believing “p” and what would make “p” true or false, what can
we say about “not -p”?

Given a sentence “p”, we may either believe or disbelieve it.
Neither of these is the primary use of a sentence; the primary

136

LOGICAL WORDS AND FALSEHOOD

use is to express belief in something else. If, feeling a drop on
my nose, I say “it is raining’’, that is what may be called “primary”
assertion, in which I pay no attention to the sentence, but use
it to refer directly to something else, namely the rain. This kind
of assertion has no corresponding negative. But if you say to me
“is it raining?” and I then look out of the window, I may answer
“yes” or “no”, and the two answers are, so to speak, at the same
level. In this case I am presented first with a sentence, and after-
wards, because of the sentence, with a meteorological fact which
enables me to say “yes” or “no”. If I answer “yes”, I am not
saying “it is raining”, but “the sentence ‘it is raining’ is true”;
for what was presented to me by your question was a sentence,
not a meteorological fact. If I answer “no”, I am saying “the
sentence ‘it is raining’ is false”. This suggests that perhaps I could
interpret “it is not raining” as meaning “the sentence ‘it is raining’
is false”.

There are, however, two difficulties about such a view. The
first is that it will make it very difficult to see what we mean
by “false”; the second is that it makes it almost impossible to
understand how a sentence containing the word “not” can be
found true by observation. When, in answer to your question,
I look out of the window, I do not merely not observe that it
is raining, for I could have achieved this without looking out;
in some sense I observe that it is not raining, but what this sense
can be is obscure.

How do I know what I assert when I say “it is not raining”?
I may say: “I saw the whole sky was blue, and I know it does
not rain when the sky is blue”. But how do I know this? Because
I have often simultaneously observed facts which I could assert
in the two sentences “the sky is blue” and “it is not raining”.
So I cannot in this way explain how I come to know negative
facts.

In what sense, if any, are there negative facts, as opposed to
true sentences containing the words “not”? Let us put the matter
as follows: Imagine a person who knew everything that can be
stated without using the word “not” or some equivalent; would
such a person know the whole course of nature, or would he not?
He would know that a buttercup is yellow, but he would not
know that it is not blue. We may say that the purpose of know-
ledge is to describe the world, and that what makes a judgment

137

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

of perception true (or false) is in general something that would
still be a fact if there were no judgments in the world. The
yellowness of the buttercup may be taken to be such a fact, and
must be mentioned in a complete description of the world. But
would there be the buttercup’s not-blueness if there were no
judgments ? And must we, in a complete description of the butter-
cup, mention all the colours that it is not?

Let us consider a case where perception leads us as directly
as possible to a very simple negative judgment. Suppose you take
sugar thinking it is salt ; when you taste it you are likely to exclaim
“this is not salt”. In such a case there is a clash between idea and
sensation : you have the idea of the taste of salt, and the sensation
of the taste of sugar, and a shock of surprise because the two are
so different. Perception only gives rise to a negative judgment
when the correlative positive judgment had already been made
or considered. When you look for something lost, you say “no,
it’s not there”; after a flash of lightning you may say “I have not
heard the thunder”. If you saw an avenue of beeches with one
elm among them, you might say “that’s not a beech”. If some
one says the whole sky is blue, and you descry a cloud on the
horizon, you may say “that is not blue”. All these are very obvious
negative judgments resulting, fairly directly, from perception. Yet,
if I see that a buttercup is yellow, I hardly seem to be adding
to my knowledge by remarking that it is not blue and not red.
What, then, is meant, in the way of objective fact, by a true
negative judgment ? 1

In all spontaneous negative perceptive judgments the experience
which leads to the judgment is, in its essential core, of one and
the same kind. There is an image or idea of a sensation of a certain
sensational class, and there is a sensation of the same class but
different from that of which there was an idea. I look for blue,
and I see red; I expect the taste of salt, and I get the taste of
sugar. Here everything is positive : idea of blue, sensation of red,
experience of difference. When I say “difference” I do not mean
mere logical non-identity, such as exists (e.g.) between a colour

1 In what follows I am concerned to show that it is possible to define
the truth of negative judgments without assuming that there are negative
facts. I profess only to construct one theory which secures this result;
I do not contend that there is no alternative theory which might be
equally satisfactory.

138

LOGICAL WORDS AND FALSEHOOD

and a taste; I mean the sort of difference that is felt between
two colours. This sort of difference is a matter of degree. We can
pass from blue to red by a series of intermediate shades, each
of which is subjectively indistinguishable from the next. We can
say that between two shades of colour there is a “great” difference,
which would be meaningless if said of a colour and a taste. Two
shades of colour have a certain kind of incompatibility: when
I see blue in a certain direction, I do not simultaneously see red
in that direction. Other kinds of sensation have a similar incom-
patibility ; at any rate this is true of sensations of touch : if I feel
a given part of the body tickled, I do not simultaneously feel it
thumped.

When, as a result of perception, I say “this is not blue”, I may
be interpreted as meaning “this is a colour differing from blue”,
where “differing” is the positive relation that might be called
“dissimilarity”, not abstract non-identity. At any rate, it may be
taken that this is the fact in virtue of which my judgment is true.
We have to distinguish between what a judgment expresses and
what it states, i.e. what makes it true or false. Thus when I say
truly “this is not blue”, there is, on the subjective side, con-
sideration of “this is blue”, followed by rejection, while on the
objective side there is some colour differing from blue. In this
way, so far as colour judgments are concerned, we escape the
need of negative facts as what make negative judgments true.

But there remains a difficulty, and a very serious one. The
above theory only succeeds in virtue of the incompatibility of
different colours, i.e. of the fact that if I see red in a given
direction I do not simultaneously see blue in that direction. This
reintroduces “not”, which we were trying to get rid of. If I
could see both blue and red simultaneously in a given direction,
then “this is red” would not be a ground for “this is not blue”.
The impossibility of seeing two colours simultaneously in a given
direction feels like a logical impossibility, not like an induction
from experience; but this is only one of various hypotheses that
are prima fade possible. Suppose that, in a given direction from
my eye, there were a source of red light, and also a source of blue
light directly behind it; I should then have some colour sensation,
which might not be either red or blue, but would be of some
single shade of colour. It would seem that the different shades
of colour are the only sensations of their kind that are physiolo-

139

HUMAN knowledge: its scope and limits

gically possible, and that there is nothing analogous to hearing
a chord in music.

Let us examine the hypothesis that the incompatibility of red
and blue is logical, and ask ourselves whether this helps us in
eliminating “not” from the objective world. We are now sup-
posing it a tautology to say: “if there is red at a given moment
in a given direction in the visual field, there is not blue in that
direction at that time”. More simply, though less accurately, we
may state our supposition as saying: “It is logically impossible
that ‘this is red 1 and ‘this is blue’ should both be true of a given
‘this’.” But this supposition, whether true or false, will not help
us. Two positive predicates, as Leibniz pointed out in proving
that God is possible, cannot be logically incompatible. Therefore
our supposition requires us to regard either “red” or “blue” or
both as complex, and one at least must contain a “not” in its
definition. For, given two complex predicates P and Q, they will
only be logically incompatible if one of them contains a con-
stituent A and the other contains a constituent not-A. In this
sense “healthy” and “ill” are incompatible, and so are “alive”
and “dead”. But there can never be logical incompatibility except
what is ultimately derived from the incompatibility of two pro-
positions/) and not-/>. Therefore we cannot eliminate “not” from
the objective world if we suppose red and blue to be logically
incompatible.

Let us examine more carefully the view that the incompatibility
of red and blue has a physiological source. That is to say, we are
to suppose that a stimulus of a certain kind causes a sensation
of red, while a stimulus of another kind causes a sensation of blue.
I incline to think this the best theory, but we then have to explain
the incompatibility of the two kinds of stimuli. As a matter of
physics, this incompatibility may be taken to arise from the fact
that each light-quantum has one definite amount of energy,
together with the quantum laws connecting energy and frequency.
The difficulty here is that it is not enough to say of a given light
quantum that it has such-and-such an amount of energy; we
must also be able to say that it does not have also some other
amount. This is always regarded as so self-evident that it is never
even stated. In classical physics analogous principles might have
had a logical basis, but in quantum physics the incompatibility
seems synthetic.

140

LOGICAL WORDS AND FALSEHOOD

Let us make a new start in the endeavour to eliminate negative
facts. Given a single indicative sentence, such as “this is red”,
we may have towards it two attitudes, belief and disbelief. Both
are “positive” in the sense that they are actual states of the
organism, which can be described without the word “not”. Each
is capable of being “true”, but the “truth” of a disbelief is not
quite the same thing as that of a belief. We considered in a
previous Chapter what is meant by the “truth” of a perceptive
belief: “this is red” is “true” if it is caused by something red.
We did not then define what makes the corresponding disbelief
“true”. Let us now address ourselves to this question.

If disbelief in “this is red” is a judgment of perception — which
is the case that we are considering — then “this” must be a colour.
It is only in logic or philosophy that we are concerned to dis-
believe in the redness of smells or sounds, and such disbelief
belongs to a later stage than that which has to be considered in
relation to our present problem. I shall therefore assume that when,
as a judgment of perception, we disbelieve “this is red”, we are
always perceiving that it is some other colour. We may therefore
say that a disbelief in “this is red” is “true” when it is caused
by something having to red the relation of positive dissimilarity
which we considered earlier. (This is a sufficient, not a necessary,
condition.)

We must now interpret the law of contradiction. We must not
say “ ‘this is red’ and ‘this is not red’ cannot both be true”, since
we are concerned to eliminate “not”. We must say “A disbelief
in the sentence ‘the belief that this is red and the disbelief that
this is red are both true' is always true”. It seems that in this
way we can replace “not” and “falsehood” by “disbelief” and
“the truth of a disbelief”. We then reintroduce “not” and “false-
hood” by definitions: the words “this is not blue” are defined
as expressing disbelief in what is expressed by the words “this
is blue”. In this way the need of “not” as an indefinable con-
stituent of facts is avoided.

The above theory may be summarized as follows: As a matter
of logic, if any propositions containing the word “not” are known,
there must be, among uninferred propositions, some that are of
the form “not -/>” or of the form “ p implies not-j”. It seems that
a judgment “this is not red” may be a judgment of perception,
provided “this” is a colour other than red. The judgment may

141

HUMAN knowledge: its scope and limits

be interpreted as disbelief in “this is red”, disbelief being a state
just as positive as belief. A sufficient (not necessary) condition
for the truth of disbelief in “this is red” is that the disbelief
should be caused by a “this” having to red the relation of positive
dissimilarity.

There is another sufficient, not necessary, test of truth in certain
cases. “This is blue” is “true” if a person whose belief is expressed
by these words will, in suitable circumstances, have a “quite-so”
feeling, and is “false” if he will get a “how-surprising” feeling.
To every belief there is a corresponding disbelief. A person
“disbelieves” what is expressed by “this is blue” if he will be
surprised if “this is blue” is true, and have a “quite-so” feeling
if “this is blue” is false. The words “this is not blue” (to repeat)
express disbelief in what is expressed by “this is blue”. Speaking
generally, “not-p” must be defined by what it expresses .

The purpose of this theory is to explain how negative sentences
can be true, and can be known, without its being necessary to
assume that there are facts which can only be asserted in sentences
containing the word “not”.

All empirical negative judgments are derived from negative
judgments of perception of the type of “this is not blue”. Suppose
you see an animal at a distance, which at first you take to be a
dog, but which, on a nearer approach, turns out to be a fox. This
depends upon perception of shape, and perception of shape
depends upon the fact that where you see one colour you do not
see another. The moment when you say “that is not a dog, but
a fox” is the moment when you see something that you did not
expect, say the fox’s brush. When your surprise is analysed it
comes down to some such judgment of perception as “this is not
green but brown”, where the fox’s brush unexpectedly hides the
grass.

There is more to be said about negation, in connection with
general propositions, and also with logic. But the above analysis
seems adequate where negative judgments of perception are con-
cerned, and generally in all cases in which observation leads us
to assert a sentence containing the word “not”.

We must now attempt a similar treatment of the word “or”.

In the case of “or” it is even more obvious than in the case
of “not” that what makes “ p or q ” true is not a fact containing
some constituent corresponding to “or”. Suppose I see an animal

142

LOGICAL WORDS AND FALSEHOOD

and say “that was a stoat or a weasel”. My statement is true if
it was a stoat, and true if it was a weasel; there is not a third
kind of animal, stoat-or- weasel. In fact, my statement expresses
partial knowledge combined with hesitation; the word “or”
expresses my hesitation, not something objective.

But it is possible to raise objections to this view. It may be said
that the word “stoat” denotes a class of animals, not all exactly
alike, and that the same is true of the word “weasel”. The phrase
“stoat Or weasel”, it may be said, merely denotes another class
of animals, which, like each of the previous classes, is composed
of individuals having common characteristics combined with
differences. There might easily be one word for stoat-or- weasel,
say “stosel”, and we could then say “that was a stosel”. This
would assert, without “or”, the same fact previously asserted with
that word.

Or, to take a simpler instance : there are many shades of blue,
having different names; there is navy-blue, aquamarine, peacock-
blue, and so on. Suppose we have a set of shades of blue, which
we will call b lt b 2) and so on, and suppose everything blue is of
one of these shades. Then the statement “this is b v or b 2 , or etc.”,
is precisely equivalent to “this is blue”, but the first statement
contains “or” while the other does not.

Such facts, however, rightly interpreted, confirm the view that
the meaning of “or” is subjective. The word “or” can be elimi-
nated without making any difference to the fact that makes a
sentence true or false, but not without making a difference to
the state of mind of a person asserting the sentence. When I say
“that is a stoat or a weasel”, I may be supposed to add “but I
don’t know which”; when I say “that is a stosel”, this addition
is absent, though it might still be true if I made it. In fact “or”
expresses conscious partial ignorance, although in logic it is
capable of other uses.

There is in this respect a difference between the standpoint
of logic and that of psychology. In logic, we are only interested
in what makes a sentence true or false ; in psychology, we are also
interested in the state of mind of the person uttering the sentence
with belief. In logic, “/>” implies “p or q” , but in psychology the
state of mind of a person asserting “p” is different from' that of
a person asserting “p or q'\ unless the person concerned is a
logician. Suppose I am asked “what day was it you went to

143

HUMAN knowledge: its scope and limits

London?” I may reply “Tuesday or Wednesday, but I don’t
remember which”. If I know that it was Tuesday, I shall not
reply “Tuesday or Wednesday”, in spite of the fact that this
answer would be true. In fact we only employ the word “or”
when we are uncertain, and if we were omniscient we should
express our knowledge without the use of this word — except,
indeed, our knowledge as to the state of mind of those aware
of a greater or less degree of ignorance.

The elimination of disjunctive “facts” is not so difficult as the
elimination of negative “facts”. It is obvious that, although I may
believe truly that to-day is Tuesday or Wednesday, there is not,
in addition to Tuesday and Wednesday, another day of the week,
called “Tuesday-or-Wednesday”. What I believe is true because
to-day is Tuesday, or because to-day is Wednesday. Here “or”
appears again, and it is true that we cannot define “or”. But
what we cannot define is not a characteristic of the non-cognitive
world, but a form of partial cognition.

Someone might argue: “When I believe ‘p or q 9 I am clearly
believing something , and this something is neither *p* nor *q\
therefore there must be something objective which is w r hat I am
believing”. This argument would be fallacious. We decided that
when I am said to be believing “not -/>” I am really disbelieving
that is to say, there is a sentence not containing the word
“not”, which denotes a certain content that I may believe or
disbelieve, but when the word “not” is added the sentence no
longer expresses merely a content, but also my attitude towards
it. The case of “or” is closely analogous. If I assert “to-day is
Tuesday or Wednesday”, there are two sentences, “to-day is
Tuesday” and “to-day is Wednesday”, each of which denotes a
certain content. My disjunctive assertion expresses a state of mind
in which neither of these contents is either affirmed or denied,
but there is hesitation between the two. The word “or” makes
the sentence one which no longer denotes a single content, but
expresses a state of mind towards two contents.

When an indicative sentence is asserted, there are three things
concerned. There is the cognitive attitude of the assertor — belief,
disbelief and hesitation, in the cases so far considered; there is
the content or contents denoted by the sentence ; and there is the
fact or facts in virtue of w r hich the sentence is true or false, which
I will call the “verifier” or “falsifier” of the sentence. In the

144

LOGICAL WORDS AND FALSEHOOD

sentence “To-day is Tuesday or Wednesday”, the cognitive atti-
tude is hesitation, the contents are two, namely the significations
of “to-day is Tuesday” and “to-day is Wednesday”; the verifier
may be the fact that it is Tuesday or the fact that it is Wednesday,
or the falsifier may be that it is a different day of the week.

A sentence containing no logical words can only express belief.
If we knew all true sentences containing no logical words, and
also knew that they were all, every other true sentence could be
obtained by logical inference. A sentence not in the list would
become true by insertion of the word “not”. A sentence in which
two sentences are connected by the word “or” would be true if
either component sentence occurred in the list. A sentence in
which two sentences are connected by the word “and” would be
true if both component sentences occurred in the list. The same
kind of logical proof would be possible for sentences containing
the logical words “all” and “some”, as will be shown in the next
Chapter.

Thus if we give the name “atomic sentence” to one not con-
taining logical words, we should need, as premisses for omniscience
(a) a list of all true atomic sentences, (6) the sentence “all true
atomic sentences occur in the above list”. We could then obtain
all other true sentences by logical inference.

But the above method fails without ( b ), when we wish to estab-
lish the truth of a sentence containing the word “all” or the
falsehood of a sentence containing the word “some”. We can,
no doubt, find substitutes for ( b ) but they will all contain, as it
does, the word “all”. It seems to follow that our knowledge must
embrace premisses containing this word, or, what is equivalent,
asserting the falsehood of sentences containing the word “some”.
This brings us to the explicit consideration of the words “all”
and “some”, which will be the subject of the next Chapter.

*45

Chapter X

GENERAL KNOWLEDGE

B y “general knowledge” I mean knowledge of the truth or
falsehood of sentences containing the word “all” or the
word “some” or logical equivalents of these words. The
word “some” might be thought to involve less generality than
the word “all”, but this would be a mistake. This appears from
the fact that the negation of a some-sentence is an all-sentence,
and vice-versa. The negation of “some men are immortal” is
“all men are mortal”, and the negation of “all men are mortal”
is “some men are immortal”. Thus any person who disbelieves
a some-sentence must believe an all-sentence, and vice versa.

The same element of universality in a some-sentence appears
from a consideration of its meaning. Suppose I say “I met a
negro in the lane”. My statement is true if I met any member
of the whole class of regroes; thus the whole class is relevant,
just as much as it would be if I said “all negroes are of African
descent”. Suppose you wanted to disprove my statement, there
would be two things you could do. First, you could go through
the whole class of negroes and prove that none of them were in
the lane; secondly, you could go through the class of people I
met, and prove that none of them were black. In either case a
complete enumeration of some class is necessary.

But as a rule a class cannot be completely enumerated. No
one can enumerate the class of negroes. If it is to be possible to
enumerate the class of people I met in the lane, we must be
able to know, concerning any member of the human race, whether
or not I met him in the lane. If I know, on a basis of perception,
that I met A, B, and C, and no one else, then I must be supposed
to know the general proposition “all human beings other than
A, B, and C were not met by me”. This raises in an acute form
the question of negative judgments of perception which we
considered in the preceding Chapter. It also makes it evident
that there are difficulties in disproving some-sentences, and
correlatively in proving all-sentences.

But before considering further the truth or falsehood of such
sentences, let us first examine what they signify.

146

GENERAL KNOWLEDGE

It is clear that the sentence “all men are mortal” can be
understood by a person who is unable to give a list of all human
beings. If you understand the logical words involved, and also
the predicates “man” and “mortal”, you can fully understand
the sentence, whether or not you can know its truth. Sometimes
you can quite certainly know the truth of such a sentence although
enumeration of the class concerned is impossible; an example
is “all primes other than 2 are odd”. This of course is a tautology;
so is the statement “all widows have been married”, which is
not known by means of an enumeration of widows. In order to
understand a general sentence, only intensions need be under-
stood; the cases in which extensions are known are exceptional.

Further: when an intension is first given, enumeration of the
corresponding extension is only possible through a universal
negative. Given, e.g. that A, B, C . . . inhabit a certain village,
this only gives the extension of “inhabitant of this village” if
we know “no human being except A, B, C . . . inhabits this
village”. Thus unless a class is defined by enumeration, it can
only be enumerated by the help of some negative all-sentence
which must be supposed known.

Although, in pure logic, an all-proposition cannot be proved
except by means of premisses which are all-propositions, there
are many all-propositions which we all believe for reasons derived
from observation. Such are “dogs bark”, “men are mortal”,
“copper conducts electricity”. The conventional view is that
such propositions are inductive generalizations, which are prob-
able, but not certain, when their premisses are known. We are
supposed to know from observation “A is a dog and A barks”,
“B is a dog and B barks”, and so on; we are also supposed not
to know any proposition of the form “X is a dog and X does not
bark”. It is supposed to follow that probably all dogs bark. I
am not concerned at the moment with the validity of such infer-
ences, but only with the fact that knowledge of the principle
guaranteeing their validity, if it exists, is general knowledge,
and of a sort which cannot be based on observation. Induction,
therefore, even if valid, does not help us to understand how we
come by general knowledge.

There are three chief methods of arriving at general pro-
positions. Sometimes they are tautologies, such as “all widows
are female”; sometimes they result from induction; sometimes

147

HUMAN knowledge: its scope and limits

they are proved by complete enumeration, e.g. “everybody in
this room is male”. I shall begin by considering complete
enumeration.

From the point of view of knowledge, though not of logic,
there is an important difference between positive and negative
general propositions, namely that some general negative pro-
positions seem to result from observation as directly as “this is
not blue”, which we considered in the preceding Chapter. In
Through the Looking-Glass the king says to Alice “who do
you see coming along the road?”, and she replies “I see nobody
coming”, to which the king retorts: “What good eyes you must
have I It’s as much as I can do to see somebody by this light”.
The point, for us, is that “I see nobody” is not equivalent to
“I do not see somebody”. The latter statement is true if my
eyes are shut, and affords no evidence that there is not somebody ;
but when I say “I see nobody”, I mean “I see, but I do not see
somebody”, which is prima facie evidence that there is not some-
body. Such negative judgments are just as important as positive
judgments in building up our empirical knowledge.

Consider, for example, such a statement as “this village has
623 inhabitants”. Census officials make such statements confi-
dently on a basis of enumeration. But enumeration involves
not only 623 propositions of the form “this is a human being”,
but also an indefinite number of propositions of the form “this
is not a human being”, and finally some assurance that I have
enough such propositions to be fairly sure that no one has been
overlooked. Jenghiz Khan believed the proposition “all the
inhabitants of Merv have been killed”, but he was wrong, because
some had crept into hiding-places that he overlooked. This was
an actual source of error; another possible source would have
been if some grotesque and long-immured prisoner had been
wrongly judged by him to be a gorilla.

Suppose you are a Gestapo officer engaged in a search, and
you satisfy yourself that at a certain time a certain house contained
just five people. What is involved in arriving at this knowledge?
Whenever you perceive a human being in any part of the house,
you cause him to come to a certain room ; when you are satisfied
that none are left, you count those whom you can see, and find
that there are five of them. This requires that, in the first place,
you should have a number of judgments “I see a man in this

148

GENERAL KNOWLEDGE

direction* * and “I see in that direction something which is not
a man**. It requires in the second place the judgment “in the
process that I went through, any man in the house would have
been perceived”. This second judgment is very likely to be mis-
taken for common-sense reasons, and we may ignore it, but the
other requires examination.

When you answer “no** to such questions as “is there a man
there?** “do you hear a noise?*’ “does that hurt?” you are
asserting a universal negative, and yet your answer seems to
result as directly from perception as if you had answered “yes’*.
This must depend upon the kind of incompatibility discussed
in the previous Chapter. You are seeing something, but its shape
differs from that of a human being; your auditory consciousness
is in the state of listening, but not of hearing; in the part of the
body concerned you are feeling something other than pain. It
is only in virtue of incompatibility that a positive percept gives
rise to a universal negative: where I see blue, I can assert that
I see no shade of red, provided the area involved is sufficiently
small. Such universal negatives based on perception raise great
difficulties, but without them most of our empirical knowledge
would be impossible, including, as we have seen, everything
statistical and everything arrived at by enumerating the members
of a class defined by intension, such as “the inhabitants of this
village” or “the people now in this room”. We must therefore
somehow find a place in our theory of knowledge for universal
negatives based on perception.

I will, however, for the moment, leave this problem aside to
examine whether there are general facts y as opposed to true
general propositions; and, if general facts are rejected, what it
is that makes general propositions true, when they are true. If
this question has been decided, it may become easier to discover
how true general propositions come to be known.

Are there general facts ? We may re-state this question in
the following form: Suppose I knew the truth or falsehood of
every sentence not containing the word “all” or the word “some”
or an equivalent of either of these words; what, then, should I
not know? Would what I should not know be only something
about my knowledge and belief, or would it be something that
involves no reference to knowledge or belief? I am supposing
that I can say “Brown is here”, “Jones is here”, “Robinson is

149

human knowledge: its scope and limits

here”, but not “some men are here”, still less “exactly three
men are here” or “every man here is called ‘Brown* or ‘Jones* or
‘Robinson* And 1 am supposing that, though I know the truth
or falsehood of every sentence of a certain sort, I do not know
that my knowledge has this completeness. If I knew my list
to be complete I could infer that there are three men here, but,
as it is, I do not know that there are no others.

Let us try to make clear exactly what is involved. When the
Antarctic Continent was discovered, something became known
which had been there before anybody knew it; the knowing was
a relation between a percipient and something which was
independent of perception and generally of the existence of life.
Is there anything analogous in the case of true all-sentences
and some-sentences, e.g. “there are volcanoes in the Antarctic”?

Let us give the name “first-order omniscience’* to knowledge
of the truth or falsehood of every sentence not containing general
words. “Limited first-order omniscience” will mean similar
complete knowledge concerning all sentences of a certain form,
say the form “ x is human”. We are to inquire what is not known
to a person with first-order omniscience.

Can we say that the only thing he does not know is that his
knowledge has first-order completeness? If so, this is a fact
about his knowledge, not about facts independent of knowledge.
It might be said that he knows everything except that there is
nothing more to know; it would seem that no fact independent
of knowing is unknown to him.

Let us take a case of limited first-order omniscience. Consider
sentences of the form “ x is human” and “ x is mortal”, and let
us suppose that a certain wise man knows whether these sentences
are true or false, for every value of for which the sentences
are significant, but does not know (what is in fact true) that there
are no other values of ‘ V* for which the sentences are significant.
Suppose A, B, C . . . Z are the values of ‘V* for which is
human” is true, and suppose that for each of these values “ x is
mortal” is true. Then the statements “A is mortal”, “B is mortal”
. . . “Z is mortal”, taken together, are in fact equivalent to “all
men are mortal”, that is to say, if one is true so is the other, and
vice versa. But our wise man cannot know this equivalence.
In any case, the equivalence involves the conjunction of “A is
mortal”, “B is mortal” . . . “Z is mortal”, that is to say, it

150

GENERAL KNOWLEDGE

involves a sentence built up by repeated use of the word “and”,
which is to be interpreted on the same lines as the word “or”.

The relation of “and” and “or” is peculiar. When I assert
“ p and #”, I can be regarded as asserting “/>” and asserting
“#”, so that the “and” of “/> and q ” seems unnecessary. But if
I deny “ p and #”, I am asserting “not -/) or not-#”, so that “or”
seems necessary for interpreting the falsehood of a conjunction.
Conversely, when I deny “/> or #”, I am asserting “not-/) and
not-#”, so that conjunction is needed to interpret the falsehood
of disjunction. Thus “and” and “or” are interdependent; either
can be defined in terms of the other plus “not”. In fact, “and”,
“or” and “not” can all be defined in terms of “not-/) or not-#”,
and also in terms of “not-/) and not-#”.

It is obvious that all-sentences are analogous to conjunctions,
and some-sentences to disjunctions.

Continuing with “all men are mortal”, let us allow our wise
man to understand “and” and “or” and “not”, but let us still
suppose him incapable of “some” and “all”. Let us further
suppose, as before, that A, B, C . . . Z are all the men there
are, and that our wise man knows “A is mortal and B is mortal
and . . . and Z is mortal”; but since he does not know the
word “all”, he does not know “A, B, C . . . Z are all the men
there are”. Let us call this proposition “P”. The question that
concerns us is : what, precisely, does he not know in not knowing P ?

In mathematical logic, P is interpreted as: “Whatever x may
be, either x is not human or x is A or x is B or . . . x is Z”.
Or it may be interpreted as : “Whatever x may be, the conjunction
‘ x is human and x is not A and x is not B and ... x is not Z’
is false.” Either of these is a statement about everything in the
universe, and it seems preposterous to suppose that we can know
about all the things in the universe. In the case of “all men”
there is real doubt, since there may be men on a planet of some
other star. But how about “all the men in this room”?

We will now suppose that A, B, C are all the men in this room,
that I know “A is in the room”, “B is in the room”, “C is in the
room”, that I understand “and” and “or” and “not”, but not
“all” or “some”, so that I cannot know “A and B and C are all
the men in this room”. W 7 e will call this proposition “Q”. What
do I not know in not knowing Q ?

Mathematical logic still brings in everything in the universe

15 1

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

in interpreting Q, which it enunciates in the form: “Whatever x
may be, either x is not in the room or x is not human or x is A
or x is B or x is C”; or “Whatever x may be, if x is not A and
x is not B and x is not C, then x is not human or x is not in the
room”. But in this case the logistical interpretation, however
convenient technically, seems obviously preposterous psychologic-
ally, for in order to know who is in the room I obviously need
not know anything about what is outside the room. How, then,
is Q to be interpreted ?

In practice, if I have seen A and B and C, and wish to be
sure of Q, I look in cupboards, under tables, and behind curtains,
and from time to time I say “there is no one in this part of the
room”. Theoretically, I could divide the volume of the room
into a number of smaller volumes, each just large enough to
contain a small human being; I could examine each volume,
and say “no one here” except where I found A and B and C.
In the end, we must be able to say “I have examined all parts of
this room” if we are to be justified in asserting Q.

The statement “no one here” is analogous to “this is not blue”,
which we considered in the preceding Chapter. It is not an indefi-
nitely extended conjunction: “Brown is not here and Jones is not
here and Robinson is not here and . . .”, through a catalogue of
the human race. What it does is to deny a character which is
common to places where there are human beings, and which we
assert when we say “some one is here”, say in playing hide-and-
seek. This raises no new problem. The universal is now in “I
have examined all parts of the room” or some equivalent.

The universal that we require may be stated as follows: “If I
go through a certain process, every person in the room will become
perceptible at some stage of the process.” The process must be
one that can be actually carried out ; we should never be justified
in saying “there are just three uranium atoms in this room”, but
human beings, fortunately, are never microscopic. Our universal
may be put in the form: “If I perform a certain series of acts
Ax, A 2 , . . . A n , every human being within a certain volume V will
be perceived during at least one of these acts.” This involves an
almost inextricable tangle of logical, physical, metaphysical, and
psychological elements, and as we are concerned at the moment
only with the logical elements it will be better to choose another
example to begin with.

152

GENERAL KNOWLEDGE

Let us take: “I have just heard six pips on the wireless.” This
may be interpreted as: “During a brief recent period of time, I
had exactly six closely similar auditory sensations of a certain well-
defined sort, namely the sort called ‘pips’.” I may give proper
names to each of these, say P 1# P 2 . . . . P 6 . Then I say: “P^ and
P 2 and . . . and P 6 were all the pips I heard in the period from the
time to the time J 2 .” We will call this statement “R”.

It is fairly obvious that what distinguishes R from the con-
junction “I heard Pj and I heard P 2 and . . . and I heard P 0 ” is
negative: it is the knowledge that I heard no other pips. Let us
consider this. Suppose I agree to listen for pips throughout a
period of five seconds, at the beginning and end of which you say
“now”. Immediately afterwards, you say “did you hear any pips”?
and I say “no”. This, though logically a universal, may be psycho-
logically a single negative judgment of perception, like “I don’t
see any blue sky” or “I don’t feel any rain”. In such judgments
(to repeat) we have the suggested idea of a quality and the sensa-
tion of a different quality which causes us to disbelieve the sugges-
ted idea. There is here no multiplicity of instances, but a specious
present in which one quality is present and another is felt to be
absent. We know “I did not hear pippiness” and we translate this
into “I heard no pips”. The plurality of “pips” is that of events
as opposed to qualities — a subject considered above in connection
with proper names.

We can extend such negative judgments beyond the specious
present, because there is no sharp boundary between sensation
and immediate memory, or between immediate memory and true
memory. You say “do you hear a pip”, and I reply, not by a sharp
“no”, but by a long-drawn out “no-o-o-o”. In this way my
negation can apply to a period of ten seconds or so. By immediate
memory and true memory it can have its temporal scope extended
indefinitely, so as to justify such a statement as “I watched all
night without seeing a single aeroplane”. W T hen such statements
are legitimate, we can say “between the time t l and the time t
I saw exactly six planes”, because we can divide the period into
smaller ones, in six of which we say “I saw a plane”, and in the
others we say “I saw no plane”. These various judgments are then
assembled in memory, and give rise to the enumerative judgment
“in the whole period I saw just six planes”.

If the above theory is correct, negative judgments of perception

153

HUMAN knowledge: its scope and limits

are not themselves universal: they say (e.g.) “I did not hear
pippiness”, not “I heard no pips”. The judgment “I heard no
pips” follows logically, for a pip is a complex of which pippiness
is a constituent. The inference is like that from “I saw no one”
to “I saw no processions”. A procession is a crowd of human
beings, and one man may at different times form part of many
processions, but processions cannot exist without human beings.
We can, therefore, from absence of the quality called “humanity”,
logically infer the absence of processions. In like manner, from
absence of noisiness we can infer absence of noises.

If the above theory is correct, enumerative empirical judgments
depend upon universal negative judgments logically inferable from
negative perceptive judgments concerned with single qualities,
such as “I do not see blue”. Our problem, so far as such judgments
are concerned, is therefore solved by the preceding theories as to
“not” and as to proper names.

The above, however, is only one of the ways in which we arrive
at general propositions. It is the way that is appropriate when
complete enumeration is possible, i.e. when there is some property
P of which we can say: “a ly a 2 , . . . a n are all the subjects of which
P can be truly asserted”. It is applicable in arriving at “this
village has 323 inhabitants”, or “all the inhabitants of this village
are called Jones” or “all mathematical logicians whose names
begin with Q live in the United States”. What we have been
discussing is: “what is involved in the possibility of complete
enumeration” 5 But there are multitudes of general propositions
in which we all believe although complete enumeration is either
practically or theoretically impossible. These are of two kinds,
tautologies and inductions. Of the former kind are “all pentagons
are polygons”, “all widows have had husbands”, etc. Of the latter
kind are “all men are mortal”, “all copper conducts electricity”,
etc. Something must be said about each of these kinds.

Tautologies are primarily relations between properties, not
between the things that have the properties. Pentagonality is a
property of which polygonality is a constituent; it may be defined
as polygonality plus quintuplicity. Thus whoever asserts penta-
gonality necessarily asserts polygonality at the same time. Simi-
larly “ x is a widow” means “ x had a husband who is dead”, and
therefore asserts, incidentally, “ x had a husband”. We have seen
that an element of tautology comes in when we seek to interpret

154

GENERAL KNOWLEDGE

such judgments as “I have heard no pips”. The strictly empirical
element is “I have not heard pippiness”; “pips” are defined as
“complexes of which pippiness is a constituent”. The inference
from “not pippiness” to “no pips” is thus tautological. I shall
say no more about tautological general propositions, since the
subject belongs to logic, with which we are not concerned.

It remains to consider inductive generalizations — not their
justification, but their significance, and what facts are necessary
if they are to be true.

That all men are mortal could, theoretically, be proved by the
enumerative method: some world-governing Caligula, having
made a complete census, might extirpate his subjects and then
commit suicide, exclaiming with his last breath: “Now I know
that all men are mortal”. But in the meantime we have to rely
upon less conclusive evidence. The most important question is
whether such generalizations, when not proved by complete
enumeration, are to be regarded as asserting a relation of intensions,
whether certain or probable, or only a relation of extensions. And
further : where there is a relation of intensions such as to justify
“all A is B”, must this be a logical relation making the generaliza-
tion tautological, or is there an extra-logical relation of intensions,
of which we acquire probable knowledge by induction ?

Take “copper conducts electricity”. This generalization was
arrived at inductively, and the induction consisted of two parts.
On the one hand, there were experiments with different bits of
copper; on the other hand, there were experiments with a variety
of substances, showing that, in every case that had been tested,
each element has a characteristic behaviour as regards the con-
duction of electricity. The same two stages exist in establishing
the induction “dogs bark”. On the one hand, we hear a number of
dogs barking; on the other hand, we observe that each species of
animal, if it makes a noise at all, makes a noise characteristic of
the species. But there is a further stage. The copper atom has been
found to have a certain structure, and from this structure, to-
gether with the general laws of physics, the conduction of electricity
can be inferred. If we now define copper as “what has a certain
atomic structure”, there is a relation between the intension
“copper” and the intension “conductivity”, which becomes
logical if the laws of physics are assumed. There is now, however,
a concealed induction, namely that what appears as copper by

155

human knowledge: its scope and limits

the tests that were applied before the modern theory of atomic
structure is also copper by the new definition. (This need only be
true in general, not universally.) This induction itself could,
theoretically, be replaced by deductions from the laws of physics.
The laws of physics themselves are partly tautologies, but in their
most important parts they are hypotheses that are found to explain
great numbers of subordinate inductions.

The same sort of thing may be said about “dogs bark”. From
the anatomy of a dog’s throat, as from that of any musical wind
instrument, it should be possible to infer that only certain sorts
of sounds can issue from it. We thus replace the rather narrow
inductive evidence derived from listening to dogs by the much
wider evidence upon which the theory of sound depends.

In all such cases the principle is the same. It is this: Given a
mass of phenomena, everything about them except an initial
space-time distribution follows tautologically from a small number
of general principles, which we therefore take to be true.

We are concerned at present, not with the validity of the
grounds for these general principles, but with the character of
what they assert, i.e. whether they assert relations of intension
or purely extensional relations of class-inclusion. I think we must
decide in favour of the former interpretation. When an induction
seems plausible, that is because a relation between the intensions
involved strikes us as not unlikely. “Logicians whose names
begin with Q live in the United States” may be proved by com-
plete enumeration, but will not be believed on inductive grounds,
because we can see no reason why a Frenchman named (say)
Quetelet should abandon his native country as soon as he became
interested in logic. On the other hand, “dogs bark” is readily
accepted on inductive grounds, because we expect a possible
answer to the question “what sort of noise do dogs make?” What
induction does, in suitable cases, is to make a relation of intensions
probable. It may do this even in cases where the general principle
suggested by induction turns out to be a tautology. You may
notice that i + 3 = 2 2 , i + 3 + S = 3 2 , 1 + 3 + 5 + 7 = 4 2 >
and be led to conjecture that the sum of the first n odd numbers
is always n 2 ; when you have framed this hypothesis, it is easy to
prove it deductively. How far ordinary scientific inductions, such
as “copper conducts electricity”, can be reduced to tautologies,
is a very difficult question, and a very ambiguous one. There are

156

GENERAL KNOWLEDGE

various possible definitions of “copper”, and the answer may
depend upon which of these definitions we adopt. I do not think,
however, that relations between intensions, such as justify state-
ments of the form “all A is B”, can always be reduced to tautolo-
gies. I am inclined to believe that there are such intensional
relations that are only discoverable empirically, and that are not,
either practically or theoretically, capable of logical demonstration.

It is necessary, before leaving this subject, to say something
about some-propositions, or existence-propositions as they are
called in logic. The statement “some A is B” is the negation of
“all A is not B” (i.e. “no A is B”), and “all A is B” is the negation
of “some A is not B”. Thus the truth of some-sentences is equiva-
lent to the falsehood of related all-sentences, and vice versa. We
have considered the truth of all-sentences, and what we have said
applies to the falsehood of some-sentences. Now we wish to con-
sider the truth of some-sentences, which involves the falsehood
of correlative all-sentences.

Suppose I met Mr. Jones, and I say to you “I met a man”. This
is a some-sentence: it asserts that, for some value of x , “I met x
and x is human” is true. I know that the x in question is Mr.
Jones, but you do not. What I know enables me to infer the truth
of “I met a man”. There is here a distinction of some importance.
If I know that the sentences “I met Jones” and “Jones is a man”
are true, it is a substantial inference that the sentence “I met a
man” is true. But if I know that I met Jones, and also that Jones
is a man, then I am already knowing that I met a man. To know
that the sentence “I met Jones” is true is not the same thing as to
know that I met Jones. I can know the latter, but not the former,
if I do not know English; I can know the former, but not the
latter, if I hear it pronounced by a person for whose moral
character I have the highest respect, but again I know no English.

Suppose you hear the door- bell, and you infer that there is a
caller. While you do not know who it is, you are in a certain state
of mind, in which belief and uncertainty are combined. When
you find out who it is, the element of uncertainty disappears, but
the element of belief remains, together with the new belief “it is
Jones”. Thus the inference from “a has the property P” to
“something has the property P” consists merely in isolating and
attending to a portion of the total belief expressed in asserting
“a has the property P”. I think something of the same sort may

157

HUMAN knowledge: its scope and limits

be said about all deductive inference, and that the difficulty of
such inference, when it exists, is due to the fact that we are be-
lieving that a sentence is true rather than what the sentence asserts.

The transition from sentences expressing judgments of per-
ception to some-sentences, e.g. from “there’s Jones” to “there’s
somebody”, thus offers no difficulty. But there are a number of
some- sentences in which we all believe, but which are not arrived
at in this simple way. We often know that something has the pro-
perty P, although there is no definite thing a of which we can
say “ a has the property P”. We know, for instance, that some one
was Mr. Jones’s father, but we may be unable to say who he was.
No one knows who was Napoleon Ill’s father, but we all believe
that some one was. If a bullet whizzes past you when no one is
in sight, you say “some one fired at me”. As a rule, in such cases,
you are making an inference from a general proposition. Everyone
has a father, therefore Mr. Jones has a father. If you believe that
everything has a cause, many things will be only known to you
as “the something that caused this”. Whether such generalizations
are the only source of some-sentences not directly derived from
perception, or whether, on the contrary, there must be some-
sentences among the premisses of our knowledge, is a question
which, for the present, I will leave open.

There is a school, of which Brouwer is the founder, which
holds that a some-sentence may be neither true nor false. The
stock example is “there are three consecutive 7’s in the decimal
expression of So far as this has been worked out, no three
consecutive 7’s have occurred. If they occur at a later point,
this may in time be discovered; but if they never occur,
this can never be discovered. I have discussed this question
in the “Inquiry into Meaning and Truth”, where I came to the
conclusion that such sentences are always either true or false if
they are syntactically significant. As I see no reason to change
this view, I shall refer the reader to that book for a statement of
my grounds, and I shall assume, without further argument, that
all syntactically correct sentences are either true or false.

Chapter XI

FACT, BELIEF, TRUTH, AND KNOWLEDGE

T he purpose of this chapter is to state in dogmatic form
certain conclusions which follow from previous discussions,
together with the fuller discussions of “An Inquiry into
Meaning and Truth”. More particularly, I wish to give meanings,
as definite as possible, to the four words in the title of this chapter.
I do not mean to deny that the words are susceptible of other
equally legitimate meanings, but only that the meanings which I
shall assign to them represent important concepts, which, when
understood and distinguished, are useful in many philosophical
problems, but when confused are a source of inextricable tangles.

A. FACT

“Fact”, as I intend the term, can only be defined ostensively.
Everything that there is in the world I call a “fact”. The sun is a
fact ; Caesar’s crossing of the Rubicon was a fact ; if I have tooth-
ache, my toothache is a fact. If I make a statement, my making it
is a fact, and if it is true there is a further fact in virtue of which
it is true, but not if it is false. The butcher says: “I’m sold out,
and that’s a fact”; immediately afterwards, a favoured customer
arrives, and gets a nice piece of lamb from under the counter. So
the butcher told two lies, one in saying he was sold out, and the
other in saying that his being sold out was a fact. Facts are what
make statements true or false. I should like to confine the word
“fact” to the minimum of what must be known in order that the
truth or falsehood of any statement may follow analytically from
those asserting that minimum. For example, if “Brutus was a
Roman” and “Cassius was a Roman” each assert a fact, I should
not say that “Brutus and Cassius were Romans” asserted a new
fact. We have seen that the questions whether there are negative
facts and general facts raise difficulties. These niceties, however,
are largely linguistic.

I mean by a “fact” something which is there, whether anybody
thinks so or not. If I look up a railway time-table and find that
there is a train to Edinburgh at io a.m., then, if the time-table is

*59

human knowledge: its scope and limits

correct, there is an actual train, which is a “fact”. The statement
in the time-table is itself a “fact”, whether true or false, but it
only states a fact if it is true, i.e. if there really is a train. Most
facts are independent of our volitions; that is why they are called
“hard”, “stubborn”, or “ineluctable”. Physical facts, for the
most part, are independent, not only of our volitions, but even
of our existence.

The whole of our cognitive life is, biologically considered, part
of the process of adaptation to facts. This process is one which
exists, in a greater or less degree, in all forms of life, but is not
commonly called “cognitive” until it reaches a certain level of
development. Since there is no sharp frontier anywhere between
the lowest animal and the most profound philosopher, it is evident
that we cannot say precisely at what point we pass from mere
animal behaviour to something deserving to be dignified by the
name of “ knowledge”. But at every stage there is adaptation, and
that to which the animal adapts itself is the environment of fact .

B. BELIEF

“Belief”, which we have next to consider, has an inherent and
inevitable vagueness, which is due to the continuity of mental
development from the amoeba to homo sapiens . In its most de-
veloped form, which is that most considered by philosophers, it is
displayed by the assertion of a sentence. After sniffing for a time,
you exclaim: “Good heavens! the house is on fire.” Or, when a
picnic is in contemplation, you say: “Look at those clouds: there
will be rain.” Or, in a train, you try to subdue an optimistic fellow-
passenger by observing: “Last time I did this journey we were
three hours late.” Such remarks, if you are not lying, express
beliefs. We are so accustomed to the use of words for expressing
beliefs that it may seem strange to speak of “belief” in cases
where there are no words. But it is clear that even when words
are used they are not of the essence of the matter. The smell of
burning first makes you believe that the house is on fire, and then
the words come, not as being the belief, but as a way of putting
it into a form of behaviour in which it can be communicated to
others. I am thinking, of course, of beliefs that are not very com-
plicated or refined. I believe that the angles of a polygon add up
to twice as many right angles as the figure has sides diminished by

160

FACT, BELIEF, TRUTH, AND KNOWLEDGE

four right angles, but a man would need super-human mathemati-
cal intuition to be able to believe this without words. But the
simpler kind of belief, especially when it calls for action, may be
entirely unverbalized. When you are travelling with a companion,
you may say: “We must run; the train is just going to start.” But
if you are alone you may have the same belief, and run just as
fast, without any words passing through your head.

I propose, therefore, to treat belief as something that can be
pre-intellectual, and can be displayed in the behaviour of animals.
I incline to think that, on occasion, a purely bodily state may
deserve to be called a “belief”. For example, if you walk into
your room in the dark, and someone has put a chair in an unusual
place, you may bump into it, because your body believed there
was no chair there. But the parts played by mind and body
respectively in belief are not very important to separate for our
present purposes. A belief, as I understand the term, is a certain
kind of state of body or mind or both. To avoid verbiage, I shall
call it a state of an organism, and ignore the distinction of bodily
and mental factors.

One characteristic of a belief is that it has external reference,
in the sense defined in a previous chapter. The simplest case,
which can be observed behaviouristically, is when, owing to a
conditioned reflex, the presence of A causes behaviour appropriate
to B. This covers the important case of acting on information
received: here the phrase heard is A, and what it signifies is B.
Somebody says “look out, there’s a car coming”, and you act as
you would if you saw the car. In this case you are believing what
is signified by the phrase “a car is coming”.

Any state of an organism which consists in believing something
can, theoretically, be fully described without mentioning the
something. When you believe “a car is coming”, your belief
consists in a certain state of the muscles, sense-organs, and emo-
tions, together perhaps with certain visual images. All this, and
whatever else may go to make up your belief, could, in theory,
be fully described by a psychologist and physiologist working
together, without their ever having to mention anything outside
your mind and body. Your state, when you believe that a car is
coming, will be very different in different circumstances. You
may be watching a race, and wondering whether the car on which
you have put your money will win. You may be waiting for the

161

HUMAN knowledge: its scope and limits

return of your son from captivity in the Far East. You may be
trying to escape from the police. You may be suddenly roused
from absent-mindedness while crossing the street. But although
your total state will not be the same in these various cases, there
will be something in common among them, and it is this something
which makes them all instances of the belief that a car is coming.
A belief, we may say, is a collection of states of an organism
bound together by all having, in whole or part, the same external
reference.

In an animal or a young child, believing is shown by an action
or series of actions. The beliefs of the hound about the fox are
shown by his following the scent. But in human beings, as a
result of language and of the practice of suspended reactions,
believing often becomes a more or less static condition, consisting
perhaps in pronouncing or imagining appropriate words, together
with one of the feelings that constitute different kinds of belief.
As to these, we may enumerate: first, the kind of belief that con-
sists in filling out sensations by animal inferences; second,
memory; third, expectation; fourth, the kind of belief generated
unreflectingly by testimony; and fifth, the kind of belief resulting
from conscious inference. Perhaps this list is both incomplete
and in part redundant, but certainly perception, memory, and
expectation differ as to the kinds of feeling involved. “Belief”,
therefore, is a wide generic term, and a state of believing is not
sharply separated from cognate states which would not naturally
be described as believings.

The question what it is that is believed when an organism is in
a state of believing is usually somewhat vague. The hound pur-
suing a scent is unusually definite, because his purpose is simple
and he has no doubt as to the means; but a pigeon hesitating
whether to eat out of your hand is in a much more vague and
complex condition. Where human beings are concerned, language
gives an illusory appearance of precision; a man may be able to
express his belief in a sentence, and it is then supposed that the
sentence is what he believes. But as a rule this is not the case. If
you say “look, there is Jones”, you are believing something, and
expressing your belief in words, but what you are believing has
to do with Jones, not with the name “Jones”. You may, on another
occasion, have a belief which is concerned with words. “Who is
that very distinguished man who has just come in ? That is Sir

162

FACT, BELIEF, TRUTH, AND KNOWLEDGE

Theophilus Thwackum.” In this case it is the name you want.
But as a rule in ordinary speech the words are, so to speak, trans-
parent; they are not what is believed, any more than a man is the
name by which he is called.

When words merely express a belief which is about what the
words mean, the belief indicated by the words is lacking in
precision to the degree that the meaning of the words is lacking
in precision. Outside logic and pure mathematics, there are no
words of which the meaning is precise, not even such words as
“centimetre” and “second”. Therefore even when a belief is
expressed in words having the greatest degree of precision of
which empirical words are capable, the question as to what it is
that is believed is still more or less vague.

This vagueness does not cease when a belief is what may be
called “purely verbal”, i.e. when what is believed is that a certain
sentence is true. This is the sort of belief acquired by schoolboys
whose education has been on old-fashioned lines. Consider the
difference in the schoolboy s attitude to “William the Conqueror,
1066” and “next Wednesday will be a whole holiday”. In the
former case, he knows that that is the right form of words, and
cares not a pin for their meaning; in the latter case, he acquires
a belief about next Wednesday, and cares not a pin what words
you use to generate his belief. The former belief, but not the
latter, is “purely verbal”.

If I were to say that the schoolboy is believing that the sentence
“William the Conqueror, 1066” is “true”, I should have to add
that his definition of “truth” is purely pragmatic: a sentence is
“true” if the consequences of uttering it in the presence of a
master are pleasant; if they are unpleasant, it is “false”.

Forgetting the schoolboy, and resuming our proper character
as philosophers, what do we mean when we say that a certain
sentence is “true”? I am not yet asking what is meant by “true”;
this will be our next topic. For the moment I am concerned to
point out that, however “true” may be defined, the significance
of “this sentence is true” must depend upon the significance of
the sentence, and is therefore vague in exactly the degree in which
there is vagueness in the sentence which is said to be true. We do
not therefore escape from vagueness by concentrating attention
on purely verbal beliefs.

Philosophy, like science, should realize that, while complete

163

HUMAN knowledge: its scope and limits

precision is impossible, techniques can be invented which
gradually diminish the area of vagueness or uncertainty. However
admirable our measuring apparatus may be, there will always
remain some lengths concerning which we are in doubt whether
they are greater than, less than, or equal to, a metre; but there
is no known limit to the refinements by which the number of such
doubtful lengths can be diminished. Similarly, when a belief is
expressed in words, there will always remain a band of possible
circumstances concerning which we cannot say whether they
would make the belief true or false, but the breadth of this band
can be indefinitely diminished, partly by improved verbal analysis,
partly by a more delicate technique in observation. Whether
complete precision is or is not theoretically possible depends
upon whether the physical world is discrete or continuous.

Let us now consider the case of a belief expressed in words all
of which have the greatest attainable degree of precision. Suppose,
for the sake of concreteness, that I believe the sentence: “My
height is greater than 5 ft. 8 ins. and less than 5 ft. 9 ins.” Let
us call this sentence “S”. I am not yet asking what would make
this sentence true, or what would entitle me to say that I know it;
I am asking only: “What is happening in me when I have the
belief which I express by the sentence S?” There is obviously no
one correct answer to this question. All that can be said definitely
is that I am in a state such as, if certain further things happen,
will give me a feeling which might be expressed by the words
“quite so”, and that, now, while these things have not yet happened,
I have the idea of their happening combined with the feeling
expressed by the word “yes”. I may, for instance, imagine myself
standing against a wall on which there is a scale of feet and inches,
and in imagination see the top of my head between two marks on
this scale, and towards this image I may have the feeling of assent.
We may take this as the essence of what may be called “static”
belief, as opposed to belief shown by action : static belief consists
in an idea or image combined with a yes-feeling.

C. TRUTH

I come now to the definition of “truth” and “falsehood”.
Certain things are evident. Truth is a property of beliefs, and
derivatively of sentences which express beliefs. Truth consists

164

FACT, BELIEF, TRUTH, AND KNOWLEDGE

in a certain relation between a belief and one or more facts other
than the belief. When this relation is absent, the belief is false.
A sentence may be called “true” or “false” even if no one believes
it, provided that, if it were believed, the belief would be true or
false as the case may be.

So much, I say, is evident. But what is not evident is the nature
of the relation between belief and fact that is involved, or the
definition of the possible fact that will make a given belief true, or
the meaning of “possible” in this phrase. Until these questions
are answered we have no adequate definition of “truth”.

Let us begin with the biologically earliest form of belief, which
is to be seen among animals as among men. The compresence of
two kinds of circumstance, A and B, if it has been frequent or
emotionally interesting, is apt to have the result that, when A is
sensibly present, the animal reacts as it formerly reacted to B, or
at any rate displays some part of this reaction. In some animals
this connection may be sometimes innate, and not the result of
experience. But however the connection may be brought about,
when the sensible presence of A causes acts appropriate to B, we
may say that the animal “believes” B to be in the environment,
and that the belief is “true” if B is in the environment. If you wake
a man up in the middle of the night and shout “fire!” he will
leap from his bed even if he does not yet see or smell fire. His
action is evidence of a belief which is “true” if there is fire, and
“false” otherwise. Whether his belief is true depends upon a fact
which may remain outside his experience. He may escape so fast
that he never acquires sensible evidence of the fire; he may fear
that he will be suspected of incendiarism and flee the country, with-
out ever inquiring whether there was a fire or not; nevertheless
his belief remains true if there was the fact (namely fire) which
constituted its external reference or significance, and if there was
not such a fact his belief remained false even if all his friends
assured him that there had been a fire.

The difference between a true and false belief is like that be-
tween a wife and a spinster : in the case of a true belief there is a
fact to which it has a certain relation, but in the case of a false
belief there is no such fact. To complete our definition of “truth”
and “falsehood” we need a description of the fact which would
make a given belief true, this description being one which applies
to nothing if the belief is false. Given a woman of whom we do not

165

HUMAN knowledge: its scope and limits

know whether she is married or not, we can frame a description
which will apply to her husband if she has one, and to nothing
if she is a spinster. Such a description would be: “the man who
stood beside her in a church or registry office while certain words
were pronounced”. In like manner we want a description of the
fact or facts which, if they exist, make a belief true. Such fact or
facts I call the “verifier” of the belief.

What is fundamental in this problem is the relation between
sensations and images, or, in Hume’s terminology, between
impressions and ideas. We have considered in a previous chapter
the relation of an idea to its prototype, and have seen how
“meaning” develops out of this relation. But given meaning and
syntax, we arrive at a new concept, which I call “significance”,
and which is characteristic of sentences and of complex images.
In the case of single words used in an exclamatory manner, such
as “fire!” or “murder!” meaning and significance coalesce, but
in general they are distinct. The distinction is made evident by
the fact that words must have meaning if they are to serve a
purpose, but a string of words does not necessarily have signifi-
cance. Significance is a characteristic of all sentences that are not
nonsensical, and not only of sentences in the indicative, but also
of such as are interrogative, imperative, or optative. For present
purposes, however, we may confine ourselves to sentences in the
indicative. Of these we may say that the significance consists in
the description of the fact which, if it exists, will make the sentence
true. It remains to define this description.

Let us take an illustration. Jefferson had a belief expressed in
the words: “There are mammoths in North America.” This
belief might have been true even if no one had seen one of these
mammoths ; there might, when he expressed the belief, have been
just two in an uninhabited part of the Rocky Mountains, and they
might soon afterwards have been swept by a flood down the
Colorado River into the sea. In that case, in spite of the truth of his
belief, there would have been no evidence for it. The actual
mammoths would have been facts, and would have been, in the
above sense, “verifiers” of the belief. A verifier which is not
experienced can often be described, if it has a relation known by
experience to something known by experience; it is in this way
that we understand such a phrase as “the father of Adam”, which
describes nothing. It is in this way that we understand Jefferson’s

1 66

FACT, BELIEF, TRUTH, AND KNOWLEDGE

belief about mammoths: we know the sort of facts that would
have made his belief true, that is to say, we can be in a state of
mind such that, if we had seen mammoths, we should have
exclaimed: “Yes, that’s what I was thinking of.”

The significance of a sentence results from the meanings of its
words together with the laws of syntax. Although meanings must
be derived from experience, significance need not. I know from
experience the meaning of “man” and the meaning of “wings”,
and therefore the significance of the sentence “There is a winged
man”, although I have no experience of what this sentence signi-
fies. The significance of a sentence may always be understood as
in some sense a description. When this description describes a
fact, the sentence is “true”; otherwise it is “false”.

It is important not to exaggerate the part played by convention.
So long as we are considering beliefs, not the sentences in which
they are expressed, convention plays no part at all. Suppose you
are expecting to meet some person of whom you are fond, and
whom you have not seen for some time. Your expectation may
be quite wordless, even if it is detailed and complex. You may
hope that he will be smiling, you may recall his voice, his gait, the
expression of his eyes ; your total expectation may be such as only
a good painter could express, in paint, not in words. In this case
you are expecting an experience of your own, and the truth or
falsehood of your expectation is covered by the relation of idea
and impression : your expectation is “true” if the impression, when
it comes, is such that it might have been the prototype of your
previous idea if the time-order had been reversed. This is what
we express when we say: “That is what I expected to see.” Con-
vention is concerned only in the translation of belief into language,
or (if we are told something) of language into belief. Moreover
the correspondence of language and belief, except in abstract
matters, is usually by no means exact: the belief is richer in detail
and context than the sentence, which picks out only certain
salient features. You say “I shall see him soon”, but you think “I
shall see him smiling, but looking older, friendly, but shy, with
his hair untidy and his shoes muddy” — and so on, through
an endless variety of detail of which you may be only half
aware.

The case of an expectation is the simplest from the point of
view of defining truth and falsehood, for in this case the fact

167

HUMAN knowledge: its scope and limits

upon which truth or falsehood depends is about to be experienced.
Other cases are more difficult.

Memory, from the standpoint of our present problem, is closely
analogous to expectation. A recollection is an idea, while the fact
recollected was an impression; the memory is “true” if the re-
collection has to the fact that kind of resemblance which exists
between an idea and its prototype.

Consider, next, such a statement as “you have a toothache”.
In any belief concerning another person’s experience there may
be the same sort of extra-verbal richness that we have seen to
be frequent in regard to expectations of our own experiences;
you may, having recently had toothache, feel sympathetically the
throbbing pangs that you imagine your friend to be suffering.
Whatever wealth or paucity of imagination you may bring to bear,
it is clear that your belief is “true” in proportion as it resembles
the fact of your friend’s toothache — the resemblance being again
of the sort that can subsist between idea and prototype.

But when we pass on to something which no one experiences
or has experienced, such as the interior of the earth, or the world
before life began, both belief and truth become more abstract
than in the above cases. We must now consider what can be
meant by “truth” when the verifying fact is experienced by no
one.

Anticipating coming discussions, I shall assume that the
physical world, as it is independently of perception, can be known
to have a certain structural similarity to the world of our percepts,
but cannot be known to have any qualitative similarity. And when
I say that it has structural similarity, I am assuming that the
ordering relations in terms of which the structure is defined are
spatio-temporal relations such as we know in our own experience.
Certain facts about the physical world, therefore — those facts,
namely, which consist of space-time structure — are such as we can
imagine. On the other hand, facts as to the qualitative character
of physical occurences are, presumably, such as we cannot
imagine.

Now while there is no difficulty in supposing that there are
unimaginable facts, there cannot be beliefs , other than general
beliefs, of which the verifiers would be unimaginable. This is an
important principle, but if it is not to lead us astray a little care is
necessary as regards certain logical points. The first of these is

1 68

FACT, BELIEF, TRUTH, AND KNOWLEDGE

that we may know a general proposition although we do not know
any instance of it. On a large pebbly beach you may say, probably
with truth: “There are pebbles on this beach which no one will
ever have noticed.” It is quite certainly true that there are finite
integers which no one will ever have thought of. But it is self-
contradictory to suppose such propositions established by giving
instances of their truth. This is only an application of the principle
that we can understand statements about all or some of the
members of a class without being able to enumerate the members.
We understand the statement “all men are mortal” just as com-
pletely as we should if we could give a complete list of men; for
to understand this statement we need only understand the concepts
“man” and “mortal” and what is meant by being an instance of
them.

Now take the statement: “There are facts which I cannot
imagine.” I am not considering whether this statement is true;
I am only concerned to show that it is intelligible. Observe, in
the first place, that if it is not intelligible, its contradictory must
also be not intelligible, and therefore not true, though also not
false. Observe, in the second place, that to understand the state-
ment it is unnecessary to be able to give instances, any more than
of the unnoticed pebbles or the numbers that are not thought of.
All that is necessary is to understand the words and the syntax,
which we do. The statement is therefore intelligible; whether it
is true is another matter.

Take, now, the following statement: “There are electrons, but
they cannot be perceived.” Again I am not asking whether the
statement is true, but what is meant by supposing it true or be-
lieving it to be true. “Electron” is a term defined by means of
causal and spatio-temporal relations to events that we experience,
and to other events related to them in ways of which we have
experience. We have experience of the relation “parent”, and can
therefore understand the relation “great-great-great-grandparent”,
although we have no experience of this relation. In like manner
we can understand sentences containing the word “electron”, in
spite of not perceiving anything to which this word is applicable.
And when I say we can understand such sentences, I mean that
we can imagine facts which would make them true.

The peculiarity, in such cases, is that we can imagine general
circumstances which would verify our belief, but cannot imagine

169

HUMAN knowledge: its scope and limits

the particular facts which are instances of the general fact. I
cannot imagine any particular fact of the form: “w is a number
which will never have been thought of”, for, whatever value 1
give to w, my statement becomes false by the very fact of my
giving that value. But I can quite well imagine the general fact
which gives truth to the statement: “There are numbers which
will never have been thought of.” The reason is that general
statements are concerned with intensions, and can be understood
without any knowledge of the corresponding extensions.

Beliefs as to what is not experienced, as the above discussion
has shown, are not as to unexperienced individuals, but as to
classes of which no member is experienced. A belief must always
be capable of being analysed into elements that experience has
made intelligible, but when a belief is set out in logical form it
often suggests a different analysis, which would seem to involve
components not known by experience. When such psychologically
misleading analysis is avoided, we can say, quite generally: Every
belief which is not merely an impulse to action is in the nature of
a picture, combined with a yes-feeling or a no-feeling; in the case
of a yes-feeling it is “true” if there is a fact having to the picture
the kind of similarity that a prototype has to an image; in the
case of a no-feeling it is “true” if there is no such fact. A belief
which is not true is called “false”.

This is a definition of “truth” and “falsehood”.

D. KNOWLEDGE

I come now to the definition of “knowledge”. As in the cases
of “belief” and “truth”, there is a certain inevitable vagueness and
inexactitude in the conception. Failure to realize this has led, it
seems to me, to important errors in the theory of knowledge.
Nevertheless, it is well to be as precise as possible about the un-
avoidable lack of precision in the definition of which we are in
search.

It is clear that knowledge is a sub-class of true beliefs: every
case of knowledge is a case of true belief, but not vice versa. It
is very easy to give examples of true beliefs that are not knowledge.
There is the man who looks at a clock which is not going, though
he thinks it is, and who happens to look at it at the moment when
it is right; this man acquires a true belief as to the time of day,

170

FACT, BELIEF, TRUTH, AND KNOWLEDGE

but cannot be said to have knowledge. There is the man who
believes, truly, that the last name of the Prime Minister in 1906
began with a B, but who believes this because he thinks that
Balfour was Prime Minister then, whereas in fact it was Campbell-
Bannerman. There is the lucky optimist who, having bought a
ticket for a lottery, has an unshakeable conviction that he will win,
and, being lucky, does win. Such instances can be multiplied
indefinitely, and show that you cannot claim to have known
merely because you turned out to be right.

What character in addition to truth must a belief have in order
to count as knowledge? The plain man would say there must be
sound evidence to support the belief. As a matter of common
sense this is right in most of the cases in which doubt arises in
practice, but if intended as a complete account of the matter it is
very inadequate. “Evidence” consists, on the one hand, of certain
matters of fact that are accepted as indubitable, and, on the other
hand, of certain principles by means of which inferences are
drawn from the matters of fact. It is obvious that this process is
unsatisfactory unless we know the matters of fact and the principles
of inference not merely by means of evidence, for otherwise we
become involved in a vicious circle or an endless regress. We
must therefore concentrate our attention on the matters of fact
and the principles of inference. We may then say that what is
known consists, first, of certain matters of fact and certain prin-
ciples of inference, neither of which stands in need of extraneous
evidence, and secondly, of all that can be ascertained by applying
the principles of inference to the matters of fact. Traditionally,
the matters of fact are those given in perception and memory,
while the principles of inference are those of deductive and in-
ductive logic.

There are various unsatisfactory features in this traditional
doctrine, though I am not at all sure that, in the end, we can
substitute anything very much better. In the first place, the
doctrine does not give an intensional definition of “knowledge”,
or at any rate not a purely intensional definition; it is not clear
what there is in common between facts of perception and prin-
ciples of inference. In the second place, as we shall see in Part III,
it is very difficult to say what are facts of perception. In the third
place, deduction has turned out to be much less powerful than was
formerly supposed ; it does not give new knowledge, except as to

171

human knowledge: its scope and limits

new forms of words for stating truths in some sense already
known. In the fourth place, the methods of inference that may
be called in a broad sense “inductive” have never been satis-
factorily formulated; when formulated, even if completely true,
they only give probability to their conclusions; moreover, in any
possibly accurate form, they lack self-evidence, and are only to be
believed, if at all, because they seem indispensable in reaching
conclusions that we all accept.

There are, broadly speaking, three ways that have been sugges-
ted for coping with the difficulties in defining “knowledge”. The
first, and oldest, is to emphasize the concept of “self-evidence”.
The second is to abolish the distinction between premisses and
conclusions, and to say that knowledge is constituted by the
coherence of a whole body of beliefs. The third and most drastic
is to abandon the concept of “knowledge” altogether and substi-
tute “beliefs that promote success” — and here “success” may
perhaps be interpreted biologically. We may take Descartes,
Hegel, and Dewey as protagonists of these three points of view.

Descartes holds that whatever I conceive clearly and distinctly
is true. He believes that, from this principle, he can derive not
only logic and metaphysics, but also matters of fact, at least in
theory. Empiricism has made such a view impossible; we do not
think that even the utmost clarity in our thoughts would enable
us to demonstrate the existence of Cape Horn. But this does not
dispose of the concept of “self-evidence”: we may say that what
he says applies to conceptual evidence, but that there is also
perceptual evidence, by means of which we come to know matters
of fact. I do not think we can entirely dispense with self-evidence.
If you slip on a piece of orange peel and hit your head with a
bump on the pavement, you will have little sympathy with a
philosopher who tries to persuade you that it is uncertain whether
you are hurt. Self-evidence also makes you accept the argument
that if all men are mortal and Socrates is a man, then Socrates is
mortal. I do not know whether self-evidence is anything except a
certain firmness of conviction ; the essence of it is that, where it is
present, we cannot help believing. If, however, self-evidence is to
be accepted as a guarantee of truth, the concept must be carefully
distinguished from others that have a subjective resemblance to
it. I think we must bear it in mind as relevant to the definition of
“knowledge”, but as not in itself sufficient.

172

FACT, BELIEF, TRUTH, AND KNOWLEDGE

Another difficulty about self-evidence is that it is a matter of
degree. A clap of thunder is indubitable, but a very faint noise is
not; that you are seeing the sun on a bright day is self-evident,
but a vague blur in a fog may be imaginary ; a syllogism in Barbara
is obvious, but a difficult step in a mathematical argument may
be very hard to “see”. It is only for the highest degree of self-
evidence that we should claim the highest degree of certainty.

The coherence theory and the instrumentalist theory are
habitually set forth by their advocates as theories of truth . As
such they are open to certain objections which I have urged else-
where. I am considering them now, not as theories of truths but
as theories of knowledge. In this form there is more to be said for
them.

Let us ignore Hegel, and set forth the coherence theory of
knowledge for ourselves. We shall have to say that sometimes two
beliefs cannot both be true, or, at least, that we sometimes believe
this. If I believe simultaneously that A is true, that B is true, and
that A and B cannot both be true, I have three beliefs which do
not form a coherent group. In that case at least one of the three
must be mistaken. The coherence theory in its extreme form
maintains that there is only one possible group of mutually co-
herent beliefs, which constitutes the whole of knowledge and the
whole of truth. I do not believe this; I hold, rather, to Leibniz’s
multiplicity of possible worlds. But in a modified form the co-
herence theory can be accepted. In this modified form it will say
that all, or nearly all, of what passes for knowledge is in a greater
or less degree uncertain ; that, if principles of inference are among
the prima facie materials of knowledge, then one piece of prima
facie knowledge may be inferrible from another, and thus acquires
more credibility than it had on its own account. It may thus
happen that a body of propositions, each of which has only a
moderate degree of credibility on its own account, may collectively
have a very high degree of credibility. But this argument depends
upon the possibility of varying degrees of intrinsic credibility, and
is therefore not a pure coherence theory. I shall consider this
matter in more detail in Part V.

With respect to the theory that we should substitute for “know-
ledge” the concept “beliefs that promote success”, it is sufficient
to point out that it derives whatever plausibility it may possess
from being half-hearted. It assumes that we can know (in the old-

173

HUMAN knowledge: its scope and limits

fashioned sense) what beliefs promote success, for if we cannot
know this the theory is useless in practice, whereas its purpose
is to glorify practice at the expense of theory. In practice, ob-
viously, it is often very difficult to know what beliefs promote
success, even if we have an adequate definition of “success”.

The conclusion to which we seem to be driven is that knowledge
is a matter of degree. The highest degree is found in facts of
perception, and in the cogency of very simple arguments. The
next highest degree is in vivid memories. When a number of
beliefs are each severally in some degree credible, they become
more so if they are found to cohere as a logical whole. General
principles of inference, whether deductive or inductive, are
usually less obvious than many of their instances, and are psycho-
logically derivative from apprehension of their instances. Towards
the end of our inquiry I shall return to the definition of “know-
ledge”, and shall then attempt to give more precision and articula-
tion to the above suggestions. Meanwhile let us remember that
the question “what do we mean by ‘knowledge’?” is not one to
which there is a definite and unambiguous answer, any more than
to the question “what do we mean by ‘baldness’?”

174

PART III

SCIENCE AND PERCEPTION

INTRODUCTION

W E come now to an inquiry which proceeds in the opposite
order from that of our initial survey of the universe. In
that survey we were attempting to be as far as possible
impartial and impersonal; it was our aim to come as near as our
capacities permit to describing the world as it might appear to an
observer of miraculous perceptive powers viewing it from without.
We were concerned with what we know rather than with what we
know. We attempted to use an order in our description which
ignored, for the moment, the fact that we are part of the universe,
and that any account which we can give of it depends upon its
effects upon ourselves, and is to this extent inevitably anthropo-
centric. We accordingly began with the system of galaxies, and
passed on, by stages, to our own galaxy, our own little solar
system, our own tiny planet, the infinitesimal specks of life upon
its surface, and finally, as the climax of insignificance, the bodies
and minds of those odd beings that have imagined themselves the
lords of creation and the end and aim of the whole vast cosmos.

But this survey, which seems to end in the pettiness of Man and
all his concerns, is only one side of the truth. There is another side,
which must be brought out by a survey of a different kind. In this
second kind of survey, which is now to occupy our attention, we
ask no longer what the universe is, but how we come to know
whatever we do know about it. In this survey Man again occupies
the centre, as in the Ptolemaic astronomy. What we know of the
world we know by means of events in our own lives, events which,
but for the power of thought, would remain merely private.
The little dots that an astronomer sees on a photographic plate are
to him signs of vast galaxies separated from him by hundreds of
thousands of light-years. All the immensities of space and all the
abysses of time are mirrored in his thought, which, in a sense,
is as vast as they are. Nothing is too great or too small for his
intellect to comprehend, nothing is too distant in time or space
for him to assign to it its due weight in the structure of the cosmos.
In power he is nearly as feeble as his minuteness suggests, but in
contemplation he is boundless, and the equal of all that he can
understand.

It is my purpose in the following Parts to discuss, first our data,

177 M

HUMAN knowledge: its scope and limits

and then the relation ofscience to the crude material of experience.
The data from which scientific inferences proceed are private to
ourselves; what we call “seeing the sun” is an event in the life of
the seer, from which the astronomer’s sun has to be inferred by a
long and elaborate process. It is evident that, if the world were a
higgledy-piggledy chaos, inferences of this kind would be im-
possible; but for causal interconnectedness, what happens in one
place would afford no indication of what has happened in another,
and my experiences would tell me nothing of events outside my
own biography. It is the process from private sensation and
thought to impersonal science that will now concern us. The road
is long and rugged, and the goal must be kept in view if the
journey is not to seem wearisome. But until we have traversed
this road neither the scope nor the essential limitations of human
knowledge can be adequately understood.

The inferences upon which we implicitly rely in this investiga-
tion, of which the explicit logic will be considered in Part VI,
differ from those of deductive logic and mathematics in being not
demonstrative, i.e. in being inferences which, when the premisses
are true and the reasoning correct, do not insure the truth of the
conclusion, though they arc held to make the conclusion “prob-
able” in some sense and in some degree. Except in mathematics,
almost all the inferences upon which we actually rely are of this
sort. In some cases the inference is so strong as to amount to
practical certainty. A page of typescript which makes sense is
assumed to have been typed by someone, although, as Eddington
points out, it may have been produced accidentally by a monkey
walking on a typewriter, and this bare possibility makes the
inference to an intentional typist non-demonstrative. Many
inferences which are accepted by all men of science are much less
nearly certain, for instance, the theory that sound is transmitted
by waves. There is a gradation in the probability assigned to
different inferences by scientific common sense, but there is no
accepted body of principles according to which such probabilities
are to be estimated. I should wish, by analysing scientific pro-
cedure, to systematize the rules of such inference. The ideal
would be the kind of systematization which has been achieved in
relation to deductive logic.

It has been customary to regard all inference as deductive or
inductive, and to regard probable inference as synonymous with

i 7 8

INTRODUCTION

inductive inference. I believe that, if ordinarily accepted scientific
inferences are to be accepted as valid, we shall have need of other
principles in addition to induction if not in place of it.

We may take three questions as typical of those that I wish to
investigate. These three are as to the best available grounds for
believing: (i) that the world existed yesterday, ( 2 ) that the sun
will rise to-morrow, (3) that there are sound-waves. I am not
asking whether these beliefs are true, but what, assuming them
true, are the best reasons for believing them. And generally: why
should we believe things asserted by science but not verified by
present perception? The answer, if I am not mistaken, is by no
means simple.

179

Chapter I

KNOWLEDGE OF FACTS
AND KNOWLEDGE OF LAWS

W HEN we examine our beliefs as to matters of fact, we find
that they are sometimes based directly on perception or
memory, while in other cases they are inferred. To com-
mon sense this distinction presents little difficulty: the beliefs that
arise immediately from perception appear to it indubitable, and the
inferences, though they may sometimes be wrong, are thought, in
such cases, to be fairly easily rectified except where peculiarly
dubious matters are concerned. I know of the existence of my
friend Mr. Jones because I see him frequently: in his presence
I know him by perception, and in his absence by memory. I know
of the existence of Napoleon because 1 have heard and read about
him, and I have every reason to believe in the veracity of my
teachers. I am somewhat less certain about Hengist and Horsa,
and much less certain about Zoroaster, but these uncertainties
are still on a common-sense level, and do not seem, at first sight,
to raise any philosophical issue.

This primitive confidence, however, was lost at a very early
stage in philosophical speculation, and was lost for sound reasons.
It was found that what I know by perception is less than has been
thought, and that the inferences by which I pass from perceived
to unperceived facts are open to question. Both these sources of
scepticism must be investigated.

There is, to begin with, a difficulty as to what is inferred and
what is not. I spoke a moment ago of my belief in Napoleon as an
inference from what I have heard and read, but there is an im-
portant sense in which this is not quite true. When a child is
being taught history, he does not argue: “My teacher is a person
of the highest moral character, paid to teach me facts; my teacher
says there was such a person as Napoleon; therefore probably
there was such a person/’ If he did, he would retain considerable
doubt, since his evidence of the teacher’s moral character is likely
to be inadequate, and in many countries at many times teachers
have been paid to teach the opposite of facts. The child in fact,
unless he hates the teacher, spontaneously believes what he is told.

180

KNOWLEDGE OF FACTS AND KNOWLEDGE OF LAWS

When we are told anything emphatically or authoritatively, it is
an effort not to believe it, as any one can experience on April
Fools* day. Nevertheless there is still a distinction, even on a
common-sense level, between what we are told and what we know
for ourselves. If you say to the child “how do you know about
Napoleon?” the child may say “because my teacher told me”.
If you say “how do you know your teacher told you”? the child
may say “why, of course, because I heard her**. If you say “how
do you know you heard her”? he may say “because I remember
it distinctly”. If you say “how do you know you remember it”?
he will either lose his temper or say “well, I do remember it”.
Until you reach this point, he will defend his belief as to a matter
of fact by belief in another matter of fact, but in the end he reaches
a belief for which he can give no further reason.

There is thus a distinction between beliefs that arise spon-
taneously and beliefs for which no further reason can be given. It
is the latter class of beliefs that are of most importance for theory
of knowledge, since they are the indispensable minimum of pre-
misses for our knowledge of matters of fact. Such beliefs I shall
call “data”. In ordinary thinking they are causes of other beliefs
rather than premisses from which other beliefs are inferred ; but
in a critical scrutiny of our beliefs as to matters of fact we must,
wherever possible, translate the causal transitions of primitive
thinking into logical transitions, and only accept the derived
beliefs to the extent that the character of the transitions seems to
justify. For this there is a common-sense reason, namely, that
every such transition is found to involve some risk of error, and
therefore data are more nearly certain than beliefs derived from
them. I am not contending that data are ever completely certain,
nor is this contention necessary for their importance in theory
of knowledge.

There is a long history of discussions as to what was mistakenly
called “scepticism of the senses”. Many appearances are deceptive.
Things seen in a mirror may be thought to be “rear*. In certain
circumstances, people see double. The rainbow seems to touch
the ground at some point, but if you go there you do not find it.
Most noteworthy in this connection are dreams: however vivid
they may have been, we believe, when we wake up, that the
objects which we thought we saw were illusory. But in all these
cases the core of data is not illusory, but only the derived beliefs.

181

HUMAN knowledge: its scope and limits

My visual sensations, when I look in a mirror or see double, are
exactly what I think they are. Things at the foot of the rainbow
do really look coloured. In dreams I have all the experiences that
I seem to have; it is only things outside my mind that are not as I
believe them to be while I am dreaming. There are in fact no
illusions of the senses, but only mistakes in interpreting sensa-
tional data as signs of things other than themselves. Or, to speak
more exactly, there is no evidence that there are illusions of the
senses.

Every sensation which is of a familiar kind brings with it
various associated beliefs and expectations. When (say) we see
and hear an aeroplane, we do not merely have a visual sensation
and the auditory sensation of a whirring noise ; spontaneously and
without conscious thought we interpret what we see and hear and
fill it out with customary adjuncts. To what an extent we do this
becomes obvious when we make a mistake, for example when what
we thought was an aeroplane turns out to be a bird. I knew a road,
along which I used often to go in a car, which had a bend at a
certain place, and a white-washed wall straight ahead. At night
it was very difficult not to see the wall as a road going straight on
up a hill. The right interpretation as a house and the wrong inter-
pretation as an up-hill road were both, in a sense, inferences from
the sensational datum, but they were not inferences in the logical
sense, since they occurred without any conscious mental process.

I give the name “animal inference” to the process of spontaneous
interpretation of sensations. When a dog hears himself called in
tones to which he is accustomed, he looks round and runs in the
direction of the sound. He may be deceived, like the dog looking
into the gramophone in the advertisement of “His Master's
Voice”. But since inferences of this sort are generated by the
repeated experiences that give rise to habit, his inference must
be one which has usually been right in his past life, since other-
wise the habit would not have been generated. We thus find
ourselves, when we begin to reflect, expecting all sorts of things
that in fact happen, although it would be logically possible for
them not to happen in spite of the occurrence of the sensations
which give rise to the expectations. Thus reflection upon animal
inference gives us an initial store of scientific laws, such as “dogs
bark”. These initial laws are usually somewhat unreliable, but
they help us to take the first steps towards science.

182

KNOWLEDGE OF FACTS AND KNOWLEDGE OF LAWS

Every-day generalizations, such as “dogs bark”, come to be
explicitly believed after habits have been generated which might
be described as a pre-verbal form of the same belief. What sort
of habit is it that comes to be expressed in the words “dogs bark” ?
We do not expect them to bark at all times, but we do expect that,
if they make a noise, it will be a bark or a growl. Psychologically,
induction does not proceed as it does in the text-books, where we
are supposed to have observed a number of occasions on which
dogs barked, and then proceeded consciously to generalize. The
fact is that the generalization, in the form of a habit of expectation,
occurs at a lower level than that of conscious thought, so that,
when we begin to think consciously, we find ourselves believing
the generalization, not, explicitly, on the basis of the evidence,
but as expressing what is implicit in our habit of expectation.
This is a history of the belief, not a justification of it.

Let us make this state of affairs somewhat more explicit. First
comes the repeated experience of dogs barking, then comes the
habit of expecting a bark, then, by giving verbal expression to
the habit, comes belief in the general proposition “dogs bark”.
Last comes the logician, who asks, not “why do I believe this”?
but “what reason is there for supposing this true”? Clearly the
reason, if any, must consist of two parts: first, the facts of per-
ception consisting of the various occasions on which we have
heard dogs bark; second, some principle justifying generalization
from observed instances to a law. But this logical process comes
historically after, not before, our belief in a host of common-sense
generalizations.

The translation of animal inferences into verbal generalizations
is carried out very inadequately in ordinary thinking, and even in
the thinking of many philosophers. In what counts as perception
of external objects there is much that consists of habits generated
by past experience. Take, for example, our belief in the perma-
nence of objects. When we see a dog or a cat, a chair or a table, we
do not suppose that we are seeing something which has a merely
momentary existence; we are convinced that what we are seeing
has a past and a future of considerable duration. We do not think
this about everything that we see ; a flash of lightning, a rocket,
or a rainbow is expected to disappear quickly. But experience has
generated in us the expectation that ordinary solid objects, which
can be touched as well as seen, usually persist, and can be seen

183

HUMAN knowledge: its scope and limits

and touched again on suitable occasions. Science reinforces this
belief by explaining away apparent disappearances as transforma-
tions into gaseous forms. But the belief in quasi-permanence,
except in exceptional cases, antedates the scientific doctrine of the
indestructibility of matter, and is itself antedated by the animal
expectation that common objects can be seen again if we look
in the right place.

The filling out of the sensational core by means of animal
inferences, until it becomes what is called “ perception”, is ana-
logous to the filling out of telegraphic press messages in news-
paper offices. The reporter telegraphs the one word “King”, and
the newspaper prints “His Gracious Majesty King George VI”.
There is some risk of error in this proceeding, since the reporter
may have been relating the doings of Mr. Mackenzie King. It is
true that the context would usually reveal such an error, but one
can imagine circumstances in which it would not. In dreams, we
fill out the bare sensational message wrongly, and only the context
of waking life shows us our mistake.

The analogy to abbreviated press telegrams is very close.
Suppose, for instance, you see a friend at the window of an in-
coming train, and a little later you see him coming towards you
on the platform. The physical causes of your perceptions (and
of your interpretation of them) are certain light-signals passing
between him and your eyes. All that physics, by itself, entitles
you to infer from the receipt of these signals is that, somewhere
along the line ol sight, light of the appropriate colours has been
emitted or reflected or refracted or scattered. It is obvious that
the kind of ingenuity which has produced the cinema could cause
you to have just these sensations in the absence of your friend, and
that in that case you would be deceived. But such sources of
deception cannot be frequent, or at least cannot have been frequent
hitherto, since, if they were, you would not have formed the
habits of expectation and belief in context that you have in fact
formed. In the case supposed, you are confident that it is your
friend, that he has existed throughout the interval between seeing
him at the window and seeing him on the platform, and that he has
pursued a continuous path through space from the one to the
other. You have no doubt that what you saw was something solid,
not an intangible object like a rainbow or a cloud. And so, although
the message received by the senses contains (so to speak) only a

184

KNOWLEDGE OF FACTS AND KNOWLEDGE OF LAWS

few key words, your mental and physical habits cause you, spon-
taneously and without thought, to expand it into a coherent and
amply informative dispatch.

This expansion of the sensational core to produce what goes
by the somewhat question-begging name of “perception” is
obviously only trustworthy in so far as our habits of association
run parallel to processes in the external world. Clouds looked
down upon from a mountain may look so like the sea or a field of
snow that only positive knowledge to the contrary prevents you
from so interpreting your visual sensations. If you are not accus-
tomed to the gramophone, you will confidently believe that the
voice you hear on the other side of the door proceeds from a
person in the room that you are about to enter. There is no
obvious limit to the invention of ingenious apparatus capable of
deceiving the unwary. We know that the people we see on the
screen in the cinema are not really there, although they move and
talk and behave in a manner having some resemblance to that of
human beings; but if we did not know it, we might at first find
it hard to believe. Thus what we seem to know through the senses
may be deceptive whenever the environment is different from
what our past experience has led us to expect.

From the above considerations it follows that we cannot admit
as data all that an uncritical acceptance of common sense would
take as given in perception. Only sensations and memories are
truly data for our knowledge of the external world. We must
exclude from our list of data not only the things that we con-
sciously infer, but all that is obtained by animal inference, such
as the imagined hardness of an object seen but not touched. It is
true that our “perceptions”, in all their fullness, are data for
psychology: we do in fact have the experience of believing in
such-and-such an object. It is only for knowledge of things outside
our own minds that it is necessary to regard only sensations as
data. This necessity is a consequence of what we know of physics
and physiology. The same external stimulus, reaching the brains
of two men with different experiences, will produce different
results, and it is only what these different results have in common
that can be used in inferring external causes. If it is objected that
the truth of physics and physiology is doubtful, the situation is
even worse; for if they are false, nothing whatever as to the outer
world can be inferred from my experiences. I am, however,

human knowledge: its scope and limits

throughout this work, assuming that science is broadly speaking
true.

If we define “data” as “those matters of fact of which, inde-
pendently of inference, we have a right to feel most nearly certain”,
it follows from what has been said that all my data are events that
happen to me, and are, in fact, what would commonly be called
events in my mind. This is a view which has been characteristic
of British empiricism, but has been rejected by most Continental
philosophers, and is not now accepted by the followers of Dewey
or by most of the logical positivists. As the issue is of considerable
importance, I shall set forth the reasons which have convinced
me, including a brief repetition of those that have already been
given.

There are, first, arguments on the common-sense level, derived
from illusions, squinting, reflection, refraction, etc., but above all
from dreams. I dreamed last night that I was in Germany, in a
house which looked out on a ruined church; in my dream I
supposed at first that the church had been bombed during the
recent war, but was subsequently informed that its destruction
dated from the wars of religion in the sixteenth century. All this,
so long as I remained asleep, had all the convincingness of waking
life. I did really have the dream, and did really have an experience
intrinsically indistinguishable from that of seeing a ruined church
when awake. It follows that the experience which I call “seeing a
church” is not conclusive evidence that there is a church, since it
may occur when there is no such external object as I suppose in
my dream. It may be said that, though when dreaming I may
think that I am awake, when I wake up I know that I am awake.
But I do not see how we are to have any such certainty; I have
frequently dreamt that I woke up; in fact once, after ether, I
dreamt it about a hundred times in the course of one dream. We
condemn dreams, in fact, because they do not fit into a proper
context, but this argument can be made inconclusive, as in
Calderon’s play, La Vida es Sueno. I do not believe that I am
now dreaming, but I cannot prove that I am not. I am, however,
quite certain that I am having certain experiences, whether they
be those of a dream or those of waking life.

We come now to another class of arguments, derived from
physics and physiology. This class of arguments came into
philosophy with Locke, who used it to show that secondary

1 86

KNOWLEDGE OF FACTS AND KNOWLEDGE OF LAWS

qualities are subjective. This class of arguments is capable of
being used to throw doubt on the truth of physics and physiology,
but I will deal with them on the hypothesis that science, in the
main, is true.

We experience a visual sensation when light waves reach the
eye, and an auditory sensation when sound waves reach the ear.
There is no reason to suppose that light waves are at all like the
experience which we call seeing something, or sound waves at all
like the experience which we call hearing a sound. There is no
reason whatever to suppose that the physical sources of light and
sound waves have any more resemblance to our experiences than
the waves have. If the waves are produced in unusual ways, our
experience may lead us to infer subsequent experiences which it
turns out that we do not have; this shows that even in normal
perception interpretation plays a larger part than common sense
supposes, and that interpretation sometimes leads us to entertain
false expectations.

Another difficulty is connected with time. We see and hear
now, but what (according to common sense) we are seeing and
hearing occurred some time ago. When we both see and hear an
explosion, we see it first and hear it afterwards. Even if we
could suppose that the furniture of our room is exactly what
it seems, we cannot suppose this of a nebula millions of light-
years away, which looks like a speck but is not much smaller
than the milky way, and of which the light that reaches us
now started before human beings began to exist. And the
difference between the nebula and the furniture is only one of
degree.

Then there are physiological arguments. People who have lost
a leg may continue to feel pain in it. Dr. Johnson, disproving
Berkeley, thought the pain in his toe when he kicked a stone was
evidence for the existence of the stone, but it appears that it was
not even evidence for the existence of his toe, since he might have
felt it even if his toe had been amputated. Speaking generally,
if a nerve is stimulated in a given manner, a certain sensation
results, whatever may be the source of the stimulation. Given
sufficient skill, it ought to be possible to make a man see the
starry heavens by tickling his optic nerve, but the instrument used
would bear little resemblance to the august bodies studied by
astronomers.

187

human knowledge: its scope and limits

The above arguments, as I remarked before, may be interpreted
sceptically, as showing that there is no reason to believe that our
sensations have external causes. As this interpretation concedes
what I am at present engaged in maintaining, namely that sensa-
tions are the sole data for physics, I shall not, for the moment,
consider whether it can be refuted, but shall pass on to a closely
similar line of argument which is related to the method of
Cartesian doubt. This method consists in searching for data by
provisionally rejecting everything that it is found possible to call
in question.

Descartes argues that the existence of sensible objects might be
uncertain, because it would be possible for a deceitful demon to
mislead us. We should substitute for a deceitful demon a cinema
in technicolour. It is, of course, also possible that we may be
dreaming. But he regards the existence of our thoughts as wholly
unquestionable. When he says “I think, therefore I am”, the
primitive certainties at which he may be supposed to have arrived
are particular “ though ts”, in the large sense in which he uses the
term. His own existence is an inference from his thoughts, an
inference whose validity does not at the moment concern us. In
the context, what appears certain to him is that there is doubting,
but the experience of doubting has no special prerogative over
other experiences. When I see a flash of lightning I may, it is
maintained, be uncertain as to the physical character of lightning,
and even as to whether anything external to myself has happened,
but I cannot make myself doubt that there has been the occurrence
which is called “seeing a flash of lightning”, though there may
have been no flash outside my seeing.

It is not suggested that I am certain about all my own experi-
ences; this would certainly be false. Many memories are dubious,
and so are many faint sensations. What I am saying — and in this
I am expounding part of Descartes’ argument — is that there are
some occurrences that I cannot make myself doubt, and that these
are all of the kind that, if we admit a not-self, are part of the life
of my self. Not all of them are sensations; some are abstract
thoughts, some are memories, some are wishes, some are pleasures
or pains. But all are what we should commonly describe as mental
events in me.

My own view is that this point of view is in the right in so far
as it is concerned with data that are matters of fact. Matters of

1 88

KNOWLEDGE OF FACTS AND KNOWLEDGE OF LAWS

tact that lie outside my experience can be made to seem doubtful,
unless there is an argument showing that their existence follows
from matters of fact within my experience together with laws of
whose certainty I feel reasonably convinced. But this is a long
question, concerning which, at the moment, I wish to say only a
few preliminary words.

Hume’s scepticism with regard to the world of science resulted
from [a) the doctrine that all my data are private to me, together
with ( b ) the discovery that matters of fact, however numerous and
well-selected, never logically imply any other matter of fact. I
do not see any way of escaping from either of these theses. The
first I have been arguing; I may say that I attach especial weight
in this respect to the argument from the physical causation of
sensations. As to the second, it is obvious as a matter of syntax to
any one who has grasped the nature of deductive arguments. A
matter of fact which is not contained in the premisses must
require for its assertion a proper name which does not occur in
the premisses. But there is only one way in which a new proper
name can occur in a deductive argument, and that is when we
proceed from the general to the particular, as m “all men are
mortal, therefore Socrates is mortar’. Now no collection of
assertions of matters of fact is logically equivalent to a general
assertion, so that, if our premisses concern only matters of fact,
this way of introducing a new proper name is not open to us.
Hence the thesis follows.

If we are not to deduce Ilume’s scepticism from the above two
premisses, there seems to be only one possible way of escape,
and that is to maintain that, among the premisses of our knowledge,
there are some general propositions, or there is at least one general
proposition, which is not analytically necessary, i.e. the hypothesis
of its falsehood is not self-contradictory. A principle justifying the
scientific use of induction would have this character. What is
needed is some way of giving probability (not certainty) to the
inferences from known matters of fact to occurrences which have
not yet been, and perhaps never will be, part of the experience
of the person making the inference. If an individual is to know
anything beyond his own experiences up to the present moment,
his stock of uninferred knowledge must consist not .only of
matters of fact, but also of general laws, or at least a law, allowing
him to make inferences from matters of fact; and such law or

189

human knowledge: its scope and limits

laws must, unlike the principles of deductive logic, be synthetic,
i.e. not proved true by their falsehood being self-contradictory.
The only alternative to this hypothesis is complete scepticism as
to all the inferences of science and common sense, including those
which I have called animal inferences.

190

Chapter II
SOLIPSISM

T he doctrine called “solipsism” is usually defined as the
belief that I alone exist. It is not one doctrine unless it is
true. If it is true, it is the assertion that 1, Bertrand Russell,
alone exist. But if it is false, and I have readers, then for you who
are reading this chapter it is the assertion that you alone exist.
This is a view suggested by the conclusions reached in the
preceding chapter, to the effect that all my data, in so far as they
are matters of fact, are private to me, and that inferences from
one or more matters of fact to other matters of fact are never
logically demonstrative. These conclusions suggest that it would
be rational to doubt everything outside my own experience, such
as the thoughts of other people and the existence of material
objects when I am not seeing them. It is this view that we are
now to examine.

We must begin by giving more precision to the doctrine, and
by distinguishing various forms that it may take. We must not
state it in the words “1 alone exist”, for these words have no clear
meaning unless the doctrine is false. If the world is really the
common-sense world of people and things, we can pick out one
person and suppose him to think that he is the whole universe.
This is analogous to the people before Columbus, who believed
the Old World to be the total of land on this planet. But if other
people and things do not exist, the word “myself” loses its mean-
ing, for this is an exclusive and delimiting word. Instead of saying
“myself is the whole universe”, we must say “data are the whole
universe”. Here “data” may be defined by enumeration. We can
then say: “this list is complete; there is nothing more”. Or we
can say: “there is not known to be anything more”. In this form,
the doctrine does not require a prior definition of the Self, and
what it asserts is sufficiently definite to be discussed.

(We may distinguish two kinds of solipsism, which I shall call
“dogmatic” and “sceptical” respectively. The dogmatic kind, in
the above statement, says “there is nothing beyond data”, while
the sceptical kind says “there is not known to be anything beyond
data”. No grounds exist in favour of the dogmatic form, since it

191

HUMAN knowledge: its scope and limits

is just as difficult to disprove existence as to prove it, when what
is concerned is something which is not a datumi I shall therefore
say no more about dogmatic solipsism, and shall concentrate on
the sceptical form.

The sceptical form of the doctrine is difficult to state precisely.
It is not right to say, as we did just now, “nothing is known
except data”, since some one else might know more; there is the
same objection as there is to dogmatic solipsism. If we emend our
statement by saying “nothing is known to me except the following
(giving a list of data)”, we have again introduced the Self, which,
as we saw, we must not do in defining our doctrine. It is not
altogether easy to evade this objection.

I think we can state the problem with which solipsism is
concerned as follows: “The propositions p x , /> 2 , . . . p n are known
otherwise than by inference. Can this list be made such that from
it other propositions, asserting matters of fact, can be inferred?”
In this form we do not have to state that our list is complete, or
that it embraces all that some one person knows.

It is obvious that if our list consists entirely of propositions
asserting matters of fact, then the answer to our question is in
the negative, and sceptical solipsism is true. But if our list contains
anything in the nature of laws the answer may be different. These
laws, however, will have to be synthetic. Any collection of matters
of fact is logically capable of being the whole ; in pure logic, any
two events are compossible, and no collection of events implies
the existence of other events.

But before pursuing this line of thought let us consider different
forms of solipsism.

Solipsism may be more drastic or less drastic; as it becomes
more drastic it becomes more logical and at the same time more
unplausible. In its least drastic form, it accepts all my mental
states that are accepted by common sense or by orthodox
psychology, i.e. not only those of which I am directly aware, but
also those that are inferred on purely psychological grounds. It is
generally held that at all times I have many faint sensations that
I do not notice. If there is a ticking clock in the room, I may
notice it and be annoyed by it, but as a rule I am quite unaware
of it, even if it is easily audible whenever I choose to listen to it.
In such a case one would naturally say that I am having auditory
sensations of which I am not conscious. The same may be said,

192

SOLIPSISM

at most times, of objects in the periphery of my field of vision.
If they are important objects, such as an enemy with a loaded
revolver, I shall quickly become aware of them and bring them
into the centre of my visual field : but if they are uninteresting and
motionless I shall remain unaware of them. Nevertheless it seems
natural to suppose that I am in some sense seeing them.

The same sort of considerations apply to lapses of memory.
If I look at an old diary, I find dinner engagements noted that I
have completely forgotten, but I find it hard to doubt that I had
the experience which common sense would describe as going to
a dinner party. I believe that I was once an infant, although no
trace of that period survives in my explicit memory.

Such inferred mental states are allowed by the least drastic
form of solipsism. It merely refuses to allow inferences to any-
thing other than myself and my mental states.

This, however, is illogical. The principles required to justify
inferences from mental states of which I am aware to others of
which I am not aware are exactly the same as those required for
inferences to physical objects and to other minds. If, therefore,
we are to secure the logical safety of which solipsism is in search,
we must confine ourselves to mental states of which we are now
aware. Buddha was admired because he could meditate while
tigers roared around him ; but if he had been a consistent solipsist
he would have held that the noise of roaring ceased as soon as he
ceased to notice it.

We thus arrive at a second form of solipsism, which says that
the universe consists, or perhaps consists, of only the following
items; and then we enumerate whatever, at the moment of
speaking, we perceive or remember. And this will have to be
confined to what I actually notice, for what I could notice is
inferred. At the moment, I notice my dog asleep, and as a plain
man I am convinced that I could have noticed him any time this
last hour, since he has been consistently (so I believe) in my field
of vision, but I have in fact been quite unaware of him. The
thoroughgoing solipsist will have to say that when, during the
last hour, my eye absent-mindedly rested on the dog, nothing
whatever occurred in me in consequence ; for to argue that I had
a sensation which I did not notice is to allow an inference of the
forbidden kind.

In regard to memory, the results of this theory are extremely

193 N

HUMAN knowledge: its scope and limits

odd. The things that I am recollecting at one moment are quite
different from those that I am recollecting at another, but the
thoroughgoing solipsist should only admit what I am remember-
ing now. Thus his world will be one of disjointed fragments
which change completely from moment to moment — change, I
mean, not as to what exists now, but as to what did exist in the
past.

But we have not done with the sacrifices which the solipsist
must make to logic if he is to feel safe. It is quite clear that I can
have a recollection without the thing remembered having hap-
pened; as a matter of logical possibility, I might have begun to
exist five minutes ago, complete with all the memories that I then
had. We ought therefore to cut out events remembered, and
confine the solipsist’s universe to present percepts, including
percepts of present states of mind which purport to be recollec-
tions. With regard to present percepts, this most rigorous type
of solipsist (if he exists) accepts the premiss of Descartes’ cogito>
with some interpretation. What he admits can only be correctly
stated in the form: “A, B, C, . . . occur.” To call A, B, C, . . .
“thoughts” adds nothing except for those who reject solipsism.
What distinguishes the consistent solipsist is the fact that the
proposition “A occurs”, if it comes in his list, is never inferred.
He rejects as invalid all inferences from one or more propositions
of the form “A occurs” to other propositions asserting the
occurrence of something, whether named or described. The
conclusions of such inferences, he maintains, may or may not
happen to be true, but can never be known to be true.

Having now stated the solipsist position, we must inquire what
can be said for and against it.

The argument for sceptical solipsism is as follows: From a
group of propositions of the form “A occurs”, it is impossible to
infer by deductive logic any other proposition asserting the
occurrence of something. If any such inference is to be valid, it
must depend upon some non-deductive principle such as causality
or induction. No such principle can be shown to be even probable
by means of deductive arguments from a group of propositions
of the form “A occurs”. (I shall be concerned in a later chapter
with the proof of this assertion.) For example, the validity of
induction cannot be inferred from the course of events except by
assuming induction or some equally questionable postulate.

194

SOLIPSISM

Therefore if, as empiricists maintain, all our knowledge is based
on experience, it must be not only based on experience, but
confined to experience ; for it is only by assuming some principle
or principles which experience cannot render even probable that
anything whatever can be proved by experience except the
experience itself.

I think this argument proves that we have to choose between
two alternatives. Either we must accept sceptical solipsism in its
most rigorous form, or we must admit that we know, indepen-
dently of experience, some principle or principles by means of
which it is possible to infer events from other events, at least with
probability. If we adopt the first alternative, we must reject far
more than solipsism is ordinarily thought to reject; we cannot
know of the existence of our own past or future, or have any
ground for expectations as to our own future, if it occurs. If we
adopt the second alternative, we must partially reject empiricism;
we must admit that we have knowledge as to certain general
features of the course of nature, and that this knowledge, though
it may be caused by experience, cannot be logically inferred from
experience. We must admit also that, if we have such knowledge,
it is not yet explicit; causality and induction, in their traditional
forms, cannot be quite true, and it is by no means clear what
should be substituted for them. It thus appears that there are
great difficulties in the way of accepting either alternative.

For my part, I reject the solipsist alternative and adopt the
other. I admit, what is of the essence of the matter, that the
solipsist alternative cannot be disproved by means of deductive
arguments, provided we grant what I shall call “the empiricist
hypothesis”, namely that what we know without inference consists
solely of what we have experienced (or, more strictly, what we are
experiencing) together with the principles of deductive logic.
But we cannot know the empiricist hypothesis to be true, since
that would be knowledge of a sort that the hypothesis itself
condemns. This does not prove the hypothesis to be false, but it
does prove that we have no right to assert it. Empiricism may be
a true philosophy, but if it is it cannot be known to be true ; those
who assert that they know it to be true contradict themselves.
There is therefore no obstacle ab initio to our rejecting the
empiricist hypothesis.

As against solipsism it is to be said, in the first place, that it is

195

human knowledge: its scope and limits

psychologically impossible to believe, and is rejected in fact even
by those who mean to accept it. I once received a letter from an
eminent logician, Mrs. Christine Ladd Franklin, saying that she
was a solipsist, and was surprised that there were no others.
Coming from a logician, this surprise surprised me. The fact that
I cannot believe something does not prove that it is false, but it
does prove that I am insincere and frivolous if I pretend to believe
it. Cartesian doubt has value as a means of articulating our
knowledge and showing what depends on what, but if carried too
far it becomes a mere technical game in which philosophy loses
seriousness. Whatever anybody, even I myself, may argue to the
contrary, I shall continue to believe that I am not the whole
universe, and in this every one will in fact agree with me, if I am
right in my conviction that other people exist.

The most important part of the argument as to solipsism is the
proof that it is only tenable in its most drastic form. There are
various half-way positions which are not altogether unplausible,
and have in fact been accepted by many philosophers. Of these
the least drastic is the view that there can never be good grounds
for asserting the existence of something which no one experiences ;
from this we may, with Berkeley, infer the unreality of matter
while retaining the reality of mind. But this view, since it admits
the experiences of others than myself, and since these experiences
are only known to me by inference, considers that it is possible
to argue validly from the existence of certain occurrences to the
existence of others; and if this is admitted, it will be found that
there is no reason why the inferred events should be experienced.
Exactly similar considerations apply to the form of solipsism
which believes that oneself has a past and a probable future ; this
belief can only be justified by admitting principles of inference
which lead to the rejection of every form of solipsism.

We are thus reduced to the two extreme hypotheses as alone
logically defensible. Either, on the one hand, we know principles
of non-deductive inference which justify our belief, not only in
other people, but in the whole physical world, including the parts
which are never perceived but only inferred from their effects;
or, on the other hand, we are confined to what may be called
“solipsism of the moment”, in which the whole of my knowledge
is limited to what I am now noticing, to the exclusion of my past
and probable future, and also of all those sensations to which, at

196

SOLIPSISM

this instant, I am not paying attention. When this alternative is
clearly realized, I do not think that anybody would honestly and
sincerely choose the second hypothesis.

If solipsism of the moment is rejected, we must seek to dis-
cover what are the synthetic principles of inference by the know-
ledge of which our scientific and common-sense beliefs are to be
justified in their broad outlines. To this task we shall address
ourselves in Part VI. But it will be well first to make a survey, on
the one hand of data, and on the other hand of scientific beliefs
interpreted in their least questionable form. By analysing the
results of this survey we may hope to discover the premisses
which, consciously or unconsciously, are assumed in the reasonings
of science.

197

Chapter III

PROBABLE INFERENCE
IN COMMON-SENSE PRACTICE

A “probable” inference (to repeat what has already been
said) is one in which, when the premisses are true and the
reasoning correct, the conclusion is nevertheless not
certain but only probable in a greater or less degree. In the
practice of science there are two kinds of inferences: those that
are purely mathematical, and those that may be called “sub-
stantial”. The inference from Kepler’s laws to the law of gravita-
tion as applied to the planets is mathematical, but the inference
from the recorded apparent motions of the planets to Kepler’s
laws is substantial, for Kepler’s laws are not the only hypotheses
logically compatible with observed facts. Mathematical inference
has been sufficiently investigated during the last half-century.
What I wish to discuss is non-mathematical inference, which is
always only probable.

I shall, broadly speaking, accept as valid any inference which
is part of the accepted body of scientific theory, unless it contains
some error of a specific kind. I shall not consider the arguments
for scepticism concerning science, but shall analyse scientific
inference on the hypothesis that it is in general valid.

My concern in this chapter will be mainly with pre-scientific
knowledge as embodied in common sense.

We must bear in mind the distinction between inference as
understood in logic and what may be called “animal inference”.
By “animal inference” I mean what happens when an occurrence
A causes a belief B without any conscious intermediary. When a
dog smells a fox he becomes excited, but we do not think that he
says to himself: “This smell has in the past been frequently
associated with the neighbourhood of a fox; therefore there is
probably a fox in the neighbourhood now.” He acts, it is true,
as he would if he went through this reasoning, but the reasoning
is performed by the body, through habit, or the “conditioned
reflex” as it is called. Whenever A has, in the animal’s past
experience, been frequently associated with B, where B is some-
thing of emotional interest, the occurrence of A tends to cause

198

PROBABLE INFERENCE IN COMMON-SENSE PRACTICE

behaviour appropriate to B. There is here no conscious connection
of A and B ; there is, we may say, A-perception and B-behaviour.
In old-fashioned language, it would be said that the “impression”
of A causes the “idea” of B. But the newer phraseology, in terms
of bodily behaviour and observable habit, is more precise and
covers a wider field.

Most substantial inferences in science, as opposed to merely
mathematical inferences, arise, in the first place, from analysis
of animal inferences. But before developing this aspect of our
subject, let us consider the scope of animal inference in human
behaviour.

The practical (as opposed to the theoretical) understanding of
language comes under the head of animal inference. Understand-
ing a word consists practically of (a) the effects of hearing it, and
( b ) the causes of uttering it. You understand the word “fox” if,
when you hear it, you have an impulse to act in a manner appro-
priate to the presence of a fox, and when you see a fox, you have
an impulse to say “fox”. But you do not need to be aware of this
connection between foxes and the word “fox”; the inference from
the word to the fox or from the fox to the word is an animal
inference. It is otherwise with erudite words, such as “dodeca-
hedron”. We learn the meaning of such words through a verbal
definition, and in such cases the connection of word and meaning
begins by being a conscious inference before it becomes a habit.

Words are a particular case of signs. We may say that, to a
given organism O, a member of a class of stimuli A is a sign 1 of
some member of a class of objects B if the occurrence to O of a
stimulus of class A produces a reaction appropriate to an object
of class B. But this is not yet quite precise. Before seeking further
precision, let us consider a concrete example, say “no smoke
without fire”.

There are various stages to be gone through before this proverb
can be enunciated. First, there must be repeated experience of
both smoke and fire, either simultaneously or in close temporal
succession. Originally, each produced its own reaction, (say)
smoke that of sniffing, and fire that of running away. But in time
a habit is formed, and smoke produces the reaction of running
away. (I am assuming an environment where forest fires are
frequent.) Some ages after the first formation of this habit, two
* Or, more correctly, a “subjective sign”.

199

human knowledge: its scope and limits

new habits are formed: smoke leads to the word “smoke”, and
fire leads to the word “fire”. Where these three habits exist—
smoke causing a reaction appropriate to fire, smoke causing the
word “smoke” and fire causing the word “fire” — the materials
exist for the formation of a fourth habit, that of the word “smoke”
causing the word “fire”. When this habit exists in a reflective
philosopher, it may cause the sentence “no smoke without fire”.
Such, at least, is a bare outline of a very complex process.

In the above example, when all these habits exist, smoke is a
sign of fire, the word “smoke” is a sign of smoke, and the word
“fire” is a sign of fire. Perhaps it may be assumed that the sign-
relation is often transitive, i.e. that, if A is a sign of B and B is
a sign of C, then A is a sign of C. This will not be invariably the
case, but it will tend to happen if the sign-relations of A and B,
B and C are very firmly established in the animal’s organism. In
that case, when the word “smoke” is a sign of smoke, and smoke
is a sign of fire, the word “smoke” will be, derivatively, a sign of
fire. If fire causes the word “fire”, the word “smoke” will thus
have become, derivatively, a cause of the word “fire”.

Let us set up a definition : An organism O has an “idea” of a
kind of object B when its action is appropriate to B although no
object of the kind B is sensibly present. This, however, requires
some limitation. An “idea” need not produce all the reactions
that would be produced by the object; this is what we mean by
saying that an idea may be faint, or not vividly imagined. There
may be nothing but the word “B”. Thus we shall say that the
idea of B is present to O whenever O shows some reaction appro-
priate to B and to nothing else.

We can now say that A is a sign of B when A causes the “idea”
of B.

We have used the word “appropriate”, and this word needs
further definition. It must not be defined teleologically, as “useful
to the organism” or what not. The reaction “appropriate” to B
is primarily the reaction caused by the sensible presence of B,
independently of acquired habits. A cry of pain on contact with
something very hot is an appropriate reaction in this sense. But
we cannot altogether exclude acquired habits from our definition
of appropriate reactions. To say “fox” when you see a fox is
appropriate. We may make a distinction: there is no situation to
which, apart from acquired habits, we react by saying “fox”.

200

PROBABLE INFERENCE IN COMMON-SENSE PRACTICE

We may therefore decide to include among “appropriate** re-
actions those which, as a result of habit, occur in the presence of
the object B, but do not occur spontaneously as reactions to
anything except B, and do not occur as habitual reactions to any-
thing other than B except as a result of a combination of habits.

The above discussion gives the definition of what may be
called a “subjective** sign, when A causes the idea of B. We may
say that A is an “objective** sign of B when A is in fact followed
or accompanied by B, and not only by the idea of B. We may say
roughly that there is error on the part of an organism whenever
a subjective sign is not also an objective sign ; but such a statement
is not correct without qualification.

Qualification is required because we must distinguish an idea
accompanied by belief from one merely entertained. If you had
two friends called Box and Cox, it is probable that the sight of
Box would cause the idea of Cox, but not the belief in the
presence of Cox. I think that entertaining an idea without belief
is a more complex occurrence than entertaining it with belief.
An idea is or involves (I will not argue which) an impulse to a
certain kind of action. When the impulse is uninhibited, the idea
is “believed”; when inhibited, the idea is merely “entertained**.
In the former case we may call the idea “active**, in the latter
“suspended**. Error is only connected with active ideas. Thus
there is error when a subjective sign produces an active idea,
although there is no such sequence between the sign and the
object of the idea.

Error, according to this view, is pre-intellectual; it requires
only bodily habits. There is error when a bird flies against a pane
of glass which it does not see. We all, like the bird, entertain rash
beliefs which may, if erroneous, lead to painful shocks. Scientific
method, I suggest, consists mainly in eliminating those beliefs
which there is positive reason to think a source of shocks, while
retaining those against which no definite argument can be brought.

In what I have been saying I have been assuming causal laws of
the form “A causes B”, where A and B are classes of occurrences.
Such laws are perhaps never wholly true. True laws can only be
expressed in differential equations. But it is not necessary that
they should be exactly true. What we need is only: “In a good
deal more than half the cases in which A occurs, B occurs simul-
taneously or soon afterwards.** This makes B probable whenever

201

HUMAN knowledge: its scope and limits

A has occurred, and that is as much as we should demand. I have
assumed that if, in the history of a given organism, A has often
been followed by B, A will be accompanied or quickly followed by
the “idea” of B, i.e. by an impulse to the actions which would be
stimulated by B. This law is inevitably vague. If A and B are
emotionally interesting to the organism, one case of their con-
junction may suffice to set up a habit; if not, many may be needed.
The conjunction of 54 and 6 times 9 has, for most children, little
emotional interest; hence the difficulty of learning the multi-
plication table. On the other hand, “once bit twice shy” shows
how easily a habit is formed when the emotional interest is strong.

As appears from what we have been saying, science starts, and
must start, from rough and ready generalizations which are only
approximately true, many of which exist as animal inferences
before they are put into words. The process is as follows: A is
followed by B a certain number of times ; then A is accompanied
by the expectation of B; then (probably much later) comes the
explicit judgment “A is a sign of B”; and only then, when
multitudes of such judgments already exist, can science begin.
Then comes Hume, with his query as to whether we ever have
reason to regard A as an objective sign of B, or even to suppose
that we shall continue to think it a sign of B. This is a sketch of
the psychology of the subject; it has no direct bearing on its logic.

The distinction between animal inference and scientific infer-
ence, I repeat, is this : In animal inference, the percept A causes
the idea of B, but there is no awareness of the connection; in
scientific inference (whether valid or invalid) there is a belief
involving both A and B, which I have expressed by “A is a sign
of B”. It is the occurrence of a single belief expressing a con-
nection of A and B that distinguishes what is commonly called
inference from what I call animal inference. But it is important
to notice that the belief expressing the connection is, in all the
most elementary cases, preceded by the habit of animal inference.

Take, as an example, the belief in more or less permanent
objects. A dog, seeing his master on different occasions, reacts in
a way which has some constant features; this is the observable
fact which we express by saying that the dog “recognizes” his
master. When the dog looks for his absent master, something
more than recognition is involved. It is difficult not to use unduly
intellectualist language to describe what occurs in such a case.

202

PROBABLE INFERENCE IN COMMON-SENSE PRACTICE

One might be tempted to say that there is a desire to replace an
idea of an object by an impression of it, but this is the sort of
phrase that seems to say much and really says little. The simplest
observable fact about desire in animals is restless behaviour until
a certain situation arises, and then relative quiescence. There are
also physiological facts about the secretions of the glands, such as
Pavlov employed. I am not denying that dogs have experiences
more or less similar to those which we have when we feel desire,
but this is an inference from their behaviour, not a datum. What
we observe can be summed up by saying that a certain part of the
dog’s behaviour is unified by reference to his master, as a planet’s
behaviour is unified by reference to the sun. In the case of the
planet, we do not infer that it “thinks” about the sun; in the case
of the dog, most of us do make the corresponding inference. But
that is a difference with which we need not be concerned as yet.

When we come to language, it is natural to have a single word
for those features of the environment which are connected
together in the kind of way in which appearances of the dog’s
master are connected together for the dog. Language has proper
names for the objects with which we are most intimately associated,
and general names for other objects. Proper names embody a
common-sense metaphysic, which, as animal inference, antedates
language. Consider such children’s questions as “Where’s
mother?” “Where’s my ball?” These imply that mothers and
balls, when not sensibly present, nevertheless exist somewhere,
and can probably be made sensible by suitable action. This belief
in permanent or quasi-permanent objects is based upon recog-
nition, and thus implies memory in some sense. However that
may be, it is clear that, by the time a child begins to speak, he
has a habit of similar reactions to a certain group of stimuli,
which, when reflected upon, becomes a belief in persistent
common-sense objects. Much the same must have been true of
mankind when developing language. The metaphysic of more or
less permanent objects underlies the vocabulary and syntax of
every language, and is the basis of the concept of substance. The
only point about it that I am concerned to make at the moment is
that it results from intellectualizing the animal inference involved
in recognition.

I come now to memory. What I wish to say about memory is,
that its general though not invariable trustworthiness is a premiss

203

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

of scientific knowledge, which is necessary if science is to be
accepted as mainly true, but is not capable of being made even
probable by arguments which do not assume memory. More
precisely : When I remember something, it is probable that what
I remember occurred, and I can form some estimate of the degree
of probability by the vividness of my recollection.

Let us first make clear what is meant logically by saying that
memory is a premiss of knowledge. It would be a mistake to set
up a general statement of the form: “What is remembered
probably occurred.” It is rather each instance of memory that is
a premiss. That is to say, we have beliefs about past occurrences
which are not inferred from other beliefs, but which, nevertheless,
we should not abandon except on very cogent grounds. (By
“we”, here, I mean people versed in scientific method and careful
as to what they believe.) The cogent grounds must necessarily
involve one or more scientific laws, and also matters of fact,
which may be either perceived or remembered. When Macbeth’s
witches vanish, he doubts whether he ever saw them, because he
believes in the persistence of material objects. But although any
memory may come to be thought mistaken, it always has a certain
weight, which makes us accept it in the absence of contrary
evidence.

A few words must be said at this point about scientific laws as
opposed to particular facts. It is only by assuming laws that one
fact can make another probable or improbable. If I remember
that at noon yesterday I was in America, but five minutes earlier
I was in Kamchatka, I shall think that one of my memories must
be mistaken, because I am firmly persuaded that the journey
cannot be performed in five minutes. But why do I think this?
As an empiricist, I hold that laws of nature should be inferred
inductively from particular facts. But how am I to establish
particular facts about how long a journey has taken ? It is clear
that I must rely partly on memory, since otherwise I shall not
know that I have taken a journey. The ultimate evidence for any
scientific law consists of particular facts, together with those
principles of scientific inference which it is my purpose to in-
vestigate. When I say that memory is a premiss, I mean that,
among the facts upon which scientific laws are based, some are
admitted solely because they are remembered. They are admitted,
however, only as probable, and any one of them may be rejected

204

PROBABLE INFERENCE IN COMMON-SENSE PRACTICE

later, after scientific laws have been discovered which make the
particular memory improbable. But this improbability is only
arrived at by assuming that most memories are veridical.

The necessity of memory as a premiss may be made evident by
asking the question : what reason have we for rejecting the hypo-
thesis that the world came into existence five minutes ago ? If it
had begun then just as, in fact, it then was, containing people
with the habits and supposed memories that in fact people then
had, there would be no possible way of finding out that they had
only just begun to exist. Yet there is nothing logically impossible
in the hypothesis. Nothing that is happening now logically implies
anything that happened at another time. And the laws of nature
by which we infer the past are themselves, as we have seen,
dependent upon memories for the evidence in their favour.
Consequently remembered facts must be included with perceived
facts as part of our data, though we may as a rule assign a lower
degree of credence to them than we do to facts of present per-
ception.

There is a distinction to be made here, which is not without
importance. A recollection is a present fact: I remember now what
I did yesterday. When I say that memory is a premiss, I do not
mean that from my present recollection I can infer the past event
recollected. This may be in some sense true, but is not the
important fact in this connection. The important fact is that the
past occurrence is itself a premiss for my knowledge. It cannot
be inferred from the present fact of my recollecting it except by
assuming the general trustworthiness of memory, i.e. that an
event remembered probably did take place. It is this that is the
memory-premiss of knowledge.

It must be understood that, when I say that this or that is a
premiss, I do not mean that it is certainly true ; I mean only that
it is something to be taken account of in arriving at the truth,
but not itself inferred from something believed to be true. The
situation is the same as that in a criminal trial in which the
witnesses contradict each other. Each witness has a certain prima
fade weight, and we have to seek a consistent system embracing
as many of their statements as possible.

I come now to another source of knowledge, namely testimony.
I do not think that the general truthfulness of testimony needs to
be a premiss in the finished structure of scientific knowledge, but

205

HUMAN knowledge: its scope and limits

it is a premiss in the early stages, and animal inference makes us
prone to believe it. Moreover I think we shall find that, in the
finished structure of science, there is a general premiss which is
needed to secure the probable trustworthiness of testimony as
well as certain other things.

Let us first consider common-sense arguments, such as would
have weight in a law court. If twelve people, each of whom lies as
often as he speaks the truth, independently testify to a certain
occurrence, the odds are 4095 to 1 that they are testifying truly.
This may be taken as practical certainty, unless the twelve people
all have a special motive for lying. This may happen. If two ships
have a collision at sea, all the sailors on one ship swear one thing,
and all the sailors on the other swear the opposite. If one of the
ships has been sunk with all hands, there will be unanimous
testimony, about which, nevertheless, lawyers experienced in
such cases will feel sceptical. But we need not pursue such argu-
ments, which are a matter for lawyers rather than philosophers.

The common-sense practice is to accept testimony unless there
is a positive reason against doing so in the particular case con-
cerned. The cause, though not the justification, of this practice
is the animal inference from a word or sentence to what it signifies.
If you are engaged in a tiger hunt and somebody exclaims
“tiger ” your body will, unless you inhibit your impulses, get
into a state very similar to that in which it would be if you saw
a tiger. Such a state is the belief that a tiger is in the neighbour-
hood; thus you will be believing the testimony of the man who
said “tiger”. The creation of such habits is half of learning the
English language; the other half is the creation of the habit of
saying “tiger ” when you see one. (I am omitting niceties of
grammar and syntax.) You can, of course, learn to inhibit the
impulse to belief ; you may come to know that your companion is
a practical joker. But an inhibited impulse still exists, and if it
ceased to exist you would cease to understand the word “tiger”.
This applies even to such dry statements as “tigers are found in
India and Eastern Asia”. You may think you hear this statement
without any of the emotions appropriate to tigers, and yet it may
cause during the following night a nightmare from which you
wake in a cold sweat, showing that the impulses appropriate to
the word “tiger” survived subconsciously.

It is this primitive credulity about testimony which causes the

206

PROBABLE INFERENCE IN COMMON-SENSE PRACTICE

success of advertising. Unless you are an unusually sophisticated
person, you will, if you are told often and emphatically that
so-and-so’s soaps or politics are the best, in the end come to
believe it, with the result that so-and-so becomes a millionaire
or a dictator as the case may be. However, I do not want to stray
into politics, so I will say no more about this aspect of the belief
in testimony.

Testimony must be distinguished from information as to the
meaning of a word, though the distinction is not always easy.
You learn the correct use of the word “cat” because your parents
say “cat” when you are noticing a cat. If they were not sufficiently
truthful for this — if, when you are noticing a cat, they said some-
times “dog”, sometimes “cow”, sometimes “crocodile” — you
could never learn to speak correctly. The fact that we do learn to
speak correctly is a testimonial to the habitual veracity of parents.
But while, from the parent’s point of view, his utterance of the
word “cat” is a statement, from the child’s point of view it is
merely a step in the acquisition of language-habits. It is only after
the child knows the meaning of the word “cat” that your utterance
of the word is a statement for him as well as for you.

Testimony is very important in one respect, namely, that it
helps to build up the distinction between the comparatively public
world of sense and the private world of thought, which is already
well established when scientific thinking begins. I was once
giving a lecture to a large audience when a cat stalked in and lay
down at my feet. The behaviour of the audience persuaded me
that I was not suffering from a hallucination. Some of our exper-
iences, but not all, appear from the behaviour of others (including
testimony) to be common to all who are in a certain neighbourhood
and are making use of normal senses. Dreams have not this public
character; no more do most “thoughts”. It must be noticed that
the public character of (say) a clap of thunder is an inference,
originally an animal inference. I hear thunder, and a person
standing beside me says “thunder”. I infer that he heard thunder,
and until I become a philosopher I make this inference with my
body, i.e. my mind emerges believing that he heard the thunder
without having gone through a “mental” process. When I become
a philosopher I have to examine the body’s inferential propensities,
including the belief in a public world which it has inferred from
observing behaviour (especially speech- behaviour) similar to its own.

207

HUMAN knowledge: its scope and limits

From the point of view of the philosopher, the interesting
question is not only whether the testimony you hear is intended
to be truthful, but whether it has any intention of conveying
information. There are here various stages towards meaningless-
ness. When you hear an actor on the stage say “I have supped full
with horrors”, you do not think he is complaining about rationing,
and you know that his statements are not intended to be believed.
When you hear a soprano voice on a phonograph lamenting her
lover’s faithlessness in anguished tones, you know that there is
no lady in the box, and that the lady who made the record was
not expressing her own emotions, but only intending to give you
pleasure in the contemplation of an imaginary sorrow. Then there
was the Scotch ghost in the eighteenth century, which kept on
repeating: ‘‘Once I was hap-hap-happy but noo I am mceserable”,
which turned out to be a rusty spit. Lastly, there are the people
in dreams, who say all sorts of things that, when we wake up, we
are convinced nobody did say.

For all these reasons, we cannot accept testimony at its face
value. The question arises : why should we accept it at all ?

We depend here, as we do when we believe in sound waves and
light waves, upon an inference going beyond our experience. Why
should not everything that seems to us to be testimony be like
either the creakings of the rusty spit or the conversation of people
in dreams? We cannot refute this hypothesis by reliance upon
experience, for our experience may be exactly the same whether
the hypothesis is true or false. And in any inference beyond future
as well as past experience we cannot rely upon induction. In-
duction argues that, if A has been frequently found to be followed
by B, it will probably be found to be followed by B next time.
This is a principle which remains entirely within experience,
actual or possible.

In the case of testimony, we depend upon analogy . The
behaviour of other people’s bodies — and especially their speech
behaviour — is noticeably similar to our own, and our own is
noticeably associated with “mental” phenomena. (For the
moment it does not matter what we mean by “mental”.) We
therefore argue that other people’s behaviour is also associated
with “mental” phenomena. Or rather, we accept this at first as
an animal inference, and invent the analogy argument afterwards
to rationalize the already existing belief.

208

PROBABLE INFERENCE IN COMMON-SENSE PRACTICE

Analogy differs from induction — at least as I am using the
words — by the fact that an analogical inference, when it passes
outside experience, cannot be verified. We cannot enter into the
minds of others to observe the thoughts and emotions which we
infer from their behaviour. We must therefore accept analogy —
in the sense in which it goes beyond experience — as an inde-
pendent premiss of scientific knowledge, or else we must find
some other equally effective principle.

The principle of analogical inference will have to be more or
less as follows : Given a class of cases in which A is accompanied
or succeeded by B, and another class of cases in which it cannot
be ascertained whether B is present or not, there is a probability
(varying according to circumstances) that in these cases also B is
present.

This is not an accurate statement of the principle, which will
need various limitations. But the necessary further refinements
would not make much difference in relation to the problem with
which we are concerned.

A further step away from experience is involved in the inference
to such things as sound waves and light waves. Let us concentrate
on the former. Suppose at a point O, from which many roads
radiate, you place a charge of gunpowder, and at a certain moment
you cause it to explode. Every hundred yards along these roads
you station an observer with a flag. A person in a stationary
balloon observes all the observers, who have orders to wave their
flags when they hear the noise of the gunpowder exploding. It is
found that those who are equidistant from O all wave their flags
at the same moment, while those who are further from O wave
their flags later than those who are nearer; moreover, the time
that elapses between the seen explosion and the waving of a given
observer's flag is proportional to his distance from O. It is
inferred by science (common sense concurring) that some process
travels outward from O, and that, therefore, something connected
with the sound is happening, not only where there are observers,
but also where there are none. In this inference we pass outside
all experience, not only outside our personal experience as in the
case of testimony. We cannot therefore interpret science wholly
in terms of experience, even when we include all experience.

The principle used in the above inference may be called,
provisionally, the principle of spatio-temporal continuity in

209 o

human knowledge: its scope and limits

causal laws. This is the same thing as denial of action at a distance.
We cannot believe that sounds arrive successively at successive
observers unless something has travelled over the intervening
space. If we deny this, our world becomes altogether too staccato
to be credible. The basis of our belief, presumably, is the con-
tinuity of all observed motions; thus perhaps analogy can be
stretched to cover this inference. However, there is much to be
said before we can be clear as to the principle governing such
inferences. I therefore leave the further consideration of this
subject for a later chapter.

So far, I have been concerned in collecting rough and ready
examples of elementary scientific inference. It remains to give
precision to the results of our preliminary survey.

I will end with a summary of the results of our present dis-
cussion.

When we begin to reflect, we find ourselves possessed of a
number of habits which may be called “animal inferences’*.
These habits consist of acting in the presence of A more or less
as we should in the presence of B, and they result from the past
conjunction of A and B in our experience. These habits, when we
become conscious of them, cause such beliefs as “A is always (or
usually) followed by B”. This is one of the main sources of the
stock of beliefs with which we start when we begin to be scientific ;
in particular, it includes the understanding of language.

Another pre-scientific belief which survives in science is the
belief in more or less permanent objects, such as people and things.
The progress of science refines this belief, and in modern quantum
theory not very much remains of it, but science could hardly have
been created without it.

The general, though not universal, trustworthiness of memory
is an independent postulate. It is necessary to much of our know-
ledge, and cannot be established by inference from anything that
does not assume it.

Testimony is, like memory, part of the sources of our primitive
beliefs. But it need not itself be made into a premiss, since it can
be merged in the wider premiss of analogy .

Finally, to infer such things as sound waves and light waves
we need a principle which may be called spatio-temporal causal
continuity, or denial of action at a distance. But this last principle
is complicated, and demands further discussion.

210

Chapter IV

PHYSICS AND EXPERIENCE

T he question to be discussed in this Chapter is one which,
in my opinion, has been far too little considered. It is this:
Assuming physics to be broadly speaking true, can we
know it to be true, and, if the answer is to be in the affirmative,
does this involve knowledge of other truths besides those of
physics? We might find that, if the world is such as physics says
it is, no organism could know it to be such ; or that, if an organism
can know it to be such, it must know some things other than
physics, more particularly certain principles of probable inference.

This question becomes acute through the problem of percep-
tion. There have, from the earliest times, been two types of theory
as to perception, one empirical, the other idealist. According to the
empirical theory, some continuous chain of causation leads from
the object to the percipient, and what is called “perceiving” the
object is the last link in this chain, or rather the last before the
chain begins to lead out of the percipient’s body instead of into
it. According to the idealist theory, when a percipient happens
to be in the neighbourhood of an object a divine illumination
causes the percipient’s soul to have an experience which is like
the object.

Each of these theories has its difficulties.

The idealist theory has its origin in Plato, but reaches its logical
culmination in Leibniz, who held that the world consists of
monads which never interact, but which all go through parallel
developments, so that what happens to me at any instant has a
similarity to what is happening to you at the same instant. When
you think you move your arm, I think I see you moving it ; thus
we are both deceived, and no one before Leibniz was sufficiently
acute to unmask the deception, which he regards as the best proof
of God’s goodness. This theory is fantastic, and has had few
adherents ; but in less logical forms portions of the idealistic theory
of perception are to be found even among those who think them-
selves most remote from it.

Philosophy is an offshoot of theology, and most philosophers,
like Malvolio, “think nobly of the soul”. They are therefore pre-

21 1

HUMAN knowledge: its scope and limits

disposed to endow it with magical powers, and to suppose that
the relation between perceiving and what is perceived must be
something utterly different from physical causation. This view
is reinforced by the belief that mind and matter are completely
disparate, and that perceiving, which is a mental phenomenon,
must be totally unlike an occurrence in the brain, which is all
that can be attributed to physical causation.

The theory that perceiving depends upon a chain of physical
causation is apt to be supplemented by a belief that to every state
of the brain a certain state of the mind “corresponds”, and vice
versa, so that, given either the state of the brain or the state of the
mind, the other could be inferred by a person who sufficiently
understood the correspondence. If it is held that there is no causal
interaction between mind and brain, this is merely a new form of
the pre-established harmony. But if causation is regarded — as it
usually is by empiricists — as nothing but invariable sequence or
concomitance, then the supposed correspondence of brain and
mind tautologically involves causal interaction. The whole
question of the dependence of mind on body or body on mind
has been involved in quite needless obscurity owing to the emo-
tions involved. The facts are quite plain. Certain observable
occurrences are commonly called “physical”, certain others
“mental”; sometimes “physical” occurrences appear as causes of
“mental” ones, sometimes vice versa. A blow causes me to feel
pain, a volition causes me to move my arm. There is no reason to
question either of these causal connections, or at any rate no
reason which does not apply to all causal connections equally.

These considerations remove one set of difficulties that stand
in the way of acceptance of the physical theory of perception.

The common-sense arguments in favour of the physical causa-
tion of perceptions are so strong that only powerful prejudices
could have caused them to be questioned. When we shut our
eyes we do not see, when we stop our ears we do not hear, when
we are under an anaesthetic we perceive nothing. The appearance
that a thing presents can be altered by jaundice, short sight,
microscopes, mists, etc. The time at which we hear a sound
depends upon our distance from its physical point of origin. The
same is true of what we see, though the velocity of light is so
great that, where terrestrial objects are concerned, the time
between an occurrence and our seeing of it is inappreciable. If it

212

PHYSICS AND EXPERIENCE

is by a divine illumination that we perceive objects, it must be
admitted that the illumination adapts itself to physical conditions.

There are, however, two objections to the physical causation
of perceptions. One is that it makes it impossible, or at least very
difficult, to suppose that external objects are what they seemto
be; the other is that it seems to make it doubtful whether the
occurrences that we call “perceptions” can really be a source of
knowledge as to the physical world. The first of these may be
ignored as having only to do with prejudices, but the second is
more important.

The problem is this : Every empiricist holds that our knowledge
as to matters of fact is derived from perception, but if physics is
true there must be so little resemblance between our percepts
and their external causes that it is difficult to see how, from per-
cepts, we can acquire a knowledge of external objects. The
problem is further complicated by the fact that physics has been
inferred from perception. Historically, physicists started from
naive realism, that is to say, from the belief that external objects
are exactly as they seem; on the basis of this assumption, they
developed a theory which made matter something quite unlike
what we perceive. Thus their conclusion contradicted their
premiss, though no one except a few philosophers noticed this.
We therefore have to decide whether, if physics is true, the hypo-
thesis of naive realism can be so modified that there shall be a
valid inference from percepts to physics. In a word : If physics is
true, is it possible that it should be known ?

Let us first try to define what we are to mean by the hypothesis
that physics is true. I want to adopt this hypothesis only to the
extent to which it appeals to educated common sense. We find
that the theories of physicists constantly undergo modification,
so that no prudent man of science would expect any physical
theory to be quite unchanged a hundred years hence. But when
theories change, the alteration usually has only a small effect so
far as observable phenomena are concerned. The practical differ-
ence between Einstein's theory of gravitation and Newton’s is
very minute, though the theoretical difference is very great.
Moreover, in every new theory there are some parts that seem
pretty certain, while others remain very speculative. Einstein’s
substitution of space-time for space and time represents a change
of language for which there are the same sort of grounds of

213

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

*■*»

simplicity as there were for the Copemican change of language.
This part of Einstein’s theory may be accepted with considerable
confidence. But the view that the universe is a three-dimensional
sphere of finite diameter remains speculative; no one would be
surprised if evidence were found which would lead astronomers
to give up this way of speaking.

Or, again, take the physical theory of light. No one doubts that
light travels at the rate of roughly 300,000 kilometers per second,
but whether it consists of waves, or of particles called photons, is
a matter as to which dispute has been possible. In the case of
sound, on the other hand, the wave theory may be accepted as
firmly established.

Every physical theory which survives goes through three stages.
In the first stage, it is a matter of controversy among specialists;
in the second stage, the specialists are agreed that it is the theory
which best fits the available evidence, though it may well here-
after be found incompatible with new evidence; in the third
stage, it is thought very unlikely that any new evidence will do
more than somewhat modify it.

When I say that I shall assume physics to be true, I mean that
I shall accept those parts of physics which have reached the third
stage, not as certain, but as more probable than any philosophical
speculation, and therefore proper to be accepted by philosophers
as a premiss in their arguments.

Let us now see what the most certain parts of physics have to
say that is relevant to our present problem.

The great physical discoveries of the seventeenth century were
made by means of two working hypotheses. One of these was that
causal laws in the physical world need only take account of matter
and motion, matter being composed of particles persisting through
time but continuously changing their positions in space. It was
assumed that, so far as physics is concerned, there is no need to
take account of anything about a particle except its position in
space at various times ; that is to say, we might suppose particles
to differ only in position, not in quality. At first, this was hardly
more than a definition of the word “physics”; when it was
necessary to take account of qualitative differences, we were
concerned with a different subject, called “chemistry”. During
the present century, however, the modern theory of the atom
has reduced chemistry, theoretically, to physics. This has enor-

214

PHYSICS AND EXPERIENCE

mously extended the scope of the hypothesis that different
particles of matter differ only in position.

Does this hypothesis apply also to physiology, or is the be-
haviour of living matter subject to laws different from those
governing dead matter? Vitalists maintain the latter view, but I
think the former has the greater weight of authority in its favour.
What can be said is that, wherever a physiological process is
understood, it is found to follow the laws of physics and chemistry,
and that, further, there is no physiological process which is clearly
not explicable by these laws. It is therefore the best hypothesis
that physiology is reducible to physics and chemistry. But this
hypothesis has not nearly the same degree of certainty as the
reduction of chemistry to physics.

I shall assume henceforth that the first of the seventeenth-
century working hypotheses, which may be called the hypothesis
of the homogeneity of matter, applies throughout the physical
world, and to living as well as dead matter. I shall not constantly
repeat that this theory is not certainly true ; this is to be taken as
said once for all. I assume the theory because the weight of
evidence, though not conclusive, seems to me strongly in its
favour.

The second of the working hypotheses of the seventeenth
century may be called the hypothesis of the independence of
causes; it is embodied in the parallelogram law. In its simplest
form it says such things as: If you walk for a minute on the deck
of a moving ship, you will reach the same point, relatively to the
water, as you would if first you stood still for a minute while the
ship moved, and then the ship stood still for a minute while you
did your walk on the deck. More generally, when a body is subject
to several forces, the result of their all acting at once for a given
length of time is the same as would be the result of their all
acting by turns, each for the given length of time — or rather, if
the given length of time is very short this will be nearly true, and
the shorter the time the more nearly true it will become. For
instance, the moon is attracted both by the earth and by the sun ;
in one second, it will move very nearly as if, for one second, it
were not attracted by either, but went on moving as before, then
for another second it were to move as if (starting from rest) it
were attracted by the earth only, then for another second as if
(starting from rest) it were attracted by the sun only. If we take

2I 5

human knowledge: its scope and limits

a shorter time than a second this will be more nearly true,
approaching the limit of complete truth as the period of time is
indefinitely diminished.

This principle is of the utmost importance technically. It
enables us, when we have studied the effects of a number of
separate forces each acting singly, to calculate the effect of their
all acting together. It is the basis of the mathematical methods
employed in traditional physics. But it must be said that it is not
self-evident, except in simple cases like that of the man walking
on the deck of the ship. In other cases, it is to be believed if it
works, but we ought not to be surprised if we find that it some-
times does not work. In the quantum theory of the atom it has had
to be abandoned, though this is perhaps not definitive. However
that may be, this second working hypothesis is much less securely
established than the first. It holds, at least approximately, over
a wide field, but there is no good ground for believing that it holds
universally.

The present century has somewhat modified the assumptions of
physics. First, there is a four-dimensional manifold of events,
instead of the two manifolds of space and time; second, causal
laws do not suffice to determine individual events, but only
statistical distributions; third, change is probably discontinuous.
These modifications would be more important to us than they are,
but for the fact that the second and third only apply effectively
to microscopic phenomena, while the physical occurrences, such
as speaking, which are associated with “mental” events, are
macroscopic. Therefore if a human body works wholly in accor-
dance with physical laws, it will still be correct to use the laws of
classical physics to determine what a man will say, and generally
what will be the large-scale motions of his body.

This brings us to the problem of the relation of mind and
matter, since perception is commonly considered “mental” while
the object perceived and the stimulus to perceiving are considered
“physical”. My own belief is that there is no difficulty whatever
about this problem. The supposed difficulties have their origin
in bad metaphysics and bad ethics. Mind and matter, we are told,
are two substances, and are utterly disparate. Mind is noble,
matter is base. Sin consists in subjection of the mind to the body.
Knowledge, being one of the noblest of mental activities, cannot
depend upon sense, for sense marks a form of subjection to

216

PHYSICS AND EXPERIENCE

matter, and is therefore bad. Hence the Platonic objection to
identifying knowledge with perception. All this, you may think,
is antiquated, but it has left a trail of prejudices hard to overcome.

Nevertheless, the distinction of mind and matter would hardly
have arisen if it had not some foundation. We must seek, therefore,
for one or more distinctions more or less analogous to the distinc-
tion between mind and matter. I should define a “mental”
occurrence as one which can be known without inference. But let
us examine some more conventional definitions.

We cannot use the Cartesian distinction between thought and
extension, if only on Leibniz’s ground, that extension involves
plurality and therefore cannot be an attribute of a single substance.
But we might try a somewhat analogous distinction. Material
things, we may say, have spatial relations, while mental things
do not. The brain is in the head, but thoughts are not — so at
least philosophers assure us. This point of view is due to a con-
fusion between different meanings of the word “space”. Among
the things that I see at a given moment there are spatial relations
which are a part of my percepts; if percepts are “mental”, as I
should contend, then spatial relations which are ingredients of
percepts are also “mental”. Naive realism identifies my percepts
with physical things; it assumes that the sun of the astronomers
is what I see. This involves identifying the spatial relations of my
percepts with those of physical things. Many people retain this
aspect of naive realism although they have rejected all the rest.

But this identification is indefensible. The spatial relations
of physics hold between electrons, protons, neutrons, etc., which
we do not perceive; the spatial relations of visual percepts hold
between things that we do perceive, and in the last analysis be-
tween coloured patches. There is a rough correlation between
physical space and visual space, but it is very rough. First: depths
become indistinguishable when they are great. Second: the
timing is different; the place where the sun seems to be now
corresponds to the place where the physical sun was eight minutes
ago. Third : the percept is subject to changes which the physicist
does not attribute to changes in the object, e.g. those brought
about by clouds, telescopes, squinting, or closing the eyes. The
correspondence between the percept and the physical 6bject is
therefore only approximate, and it is no more exact as regards
spatial relations than it is in other respects. The sun of the

217

HUMAN knowledge: its scope and limits

physicist is not identical with the sun of my percepts, and the
93,000,000 miles that separate it from the moon are not identical
with the spatial relation between the visual sun and the visual
moon when I happen to see both at once.

When I say that something is “outside” me, there are two
different things that I may mean. I may mean that I have a
percept which is outside the percept of my body in perceptual
space, or I may mean that there is a physical object which is
outside my body as a physical object in the space of physics.
Generally there is a rough correspondence between these two.
The table that I see is outside my body as I see it in perceptual
space, and the physical table is outside my physical body in
physical space. But sometimes the correspondence fails. I dream,
say, of a railway accident: I see the train falling down an em-
bankment, and I hear the shrieks of the injured. These dream-
objects are genuinely and truly “outside” my dream body in my
own perceptual space. But when I wake up I find that the whole
dream was due to a noise in my ear. And when I say that the noise
is in my ear, I mean that the physical source of the sound that I
experience is “in” my ear as a physical object in physical space.
In another sense, we might say that all noises are in the ear, but
if we confuse these two senses the result is an inextricable tangle.

Generalizing, we may say that my percept of anything other
than my body is “outside* * the percept of my body in perceptual
space, and if the perception is not misleading the physical object
is “outside** my physical body in physical space. It does not
follow that my percept is outside my physical body. Indeed, such
a hypothesis is prima facie meaningless, although, as we shall see.
a meaning can be found for it, and it is then false.

We can now begin to tackle our central question, namely, what
do we mean by a “percept**, and how can it be a source of know-
ledge as to something other than itself?

What is a “percept**? As I use the word, it is what happens
when, in common-sense terms, I see something or hear some-
thing or otherwise believe myself to become aware of something
through my senses. The sun, we believe, is always there, but I
only sometimes see it: I do not see it at night, or in cloudy
weather, or when I am otherwise occupied. But sometimes I see
it. All the occasions on which I see the sun have a certain re-
semblance to each other, which enabled me in infancy to learn

218

PHYSICS AND EXPERIENCE

to use the word “sun” on the right occasions. Some of the re-
semblances between different occasions when I see the sun are
obviously in me; for example, I must have my eyes open and turn
in the right direction. These, therefore, we do not regard as
properties of the sun. But there are other resemblances which, so
far as common sense can discover, do not depend upon us ; when
we see the sun, it is almost always round and bright and hot. The
few occasions when it is not are easily explicable as due to fog or
to an eclipse. Common sense therefore says: there is an object
which is round and bright and hot; the kind of event called
“seeing the sun” consists in a relation between me and this object,
and when this relation occurs I am “perceiving” the object.

But at this point physics intervenes in a very awkward way. It
assures us that the sun is not “bright” in the sense in which we
usually understand the word ; it is a source of light-rays which have
a certain effect upon eyes and nerves and brains, but when this
effect is absent because the light-rays do not encounter a living
organism, there is nothing that can be properly called “brightness”.
Exactly the same considerations apply to the words “hot” and
“round” — at least if “round” is understood as a perceptible
quality. Moreover, though you see the sun now, the physical
object to be inferred from your seeing existed eight minutes ago;
if, in the intervening minutes, the sun had gone out, you would
still be seeing exactly what you are seeing. We cannot therefore
identify the physical sun with what we see ; nevertheless what we
see is our chief reason for believing in the physical sun.

Assuming the truth of physics, what is there in its laws that
justifies inferences from percepts to physical objects? Before we
can adequately discuss this question, we must determine the
place of percepts in the world of physics. There is here a peculi-
arity : physics never mentions percepts except when it speaks of
empirical verification of its laws ; but if its laws are not concerned
with percepts, how can percepts verify them? This question
should be borne in mind during the following discussions.

The question of the position of percepts in the causal chains of
physics is a different one from that of the cognitive status of
percepts, though the two are interconnected. At the moment I am
concerned with the location of percepts in causal chains. Now
a percept — say hearing a noise — has a series of antecedents, which
travel in space-time from the physical source of the noise through

219

HUMAN knowledge: its scope and limits

the air to the ears and brain. The experience which we call
“hearing the noise ,, is as nearly as can be determined simultaneous
with the cerebral term of the physical causal chain. If the noise is
of the kind to call forth a bodily movement, the movement begins
almost immediately after “hearing the noise”. If we are going to
fit “hearing the noise” into a physical causal chain, we must
therefore connect it with the same region of space-time as that of
the accompanying cerebral events. And this applies also to the
noise as something perceived. The only region of space-time with
which this noise has any direct connection is the present state of
the hearer’s brain ; the connection with the physical source of the
sound is indirect. Exactly the same argument applies to things
seen.

I am anxious to minimize the metaphysical assumptions to be
made in this connection. You may hold that mind and matter
interact, or that, as the Cartesians contended, they run in parallel
series, or that, as materialists believe, mental occurrences are
mere concomitants of certain physical occurrences, determined
by them but having no reciprocal influence on physical events.
What you hold in these respects has no bearing on the point that
I am making. What I am saying is something which is obvious
to educated common sense, namely that, whether we consider the
percept or the simultaneous state of the brain, the causal location
of either is intermediate between occurrences in afferent nerves
constituting the stimulus, and occurrences in efferent nerves
constituting the reaction.

This applies not only to the perceiving, which we naturally
regard as “mental”, but to what we experience when we perceive.
That is to say, it applies not only to “seeing the sun”, but also
to the sun, if we mean by “the sun” something that a human
being can experience. The astronomer’s sun is inferred, it is not
hot or bright, and it existed eight minutes before what is called
seeing it. If I see the sun and it makes me blink, what I see is not
93,000,000 miles and eight minutes away, but is causally (and
therefore spatio-temporally) intermediate between the light-waves
striking the eye and the consequent blinking.

The dualistic view of perception, as a relation of a subject to
an object, is one which, following the leadership of William
James, empiricists have now for the most part abandoned. The
distinction between “seeing the sun” as a mental event, and the

220

PHYSICS AND EXPERIENCE

immediate object of my seeing, is now generally rejected as
invalid, and in this view I concur. But many of those who take
the view that I take on this point nevertheless inconsistently
adhere to some form of naive realism. If my seeing of the sun is
identical with the sun that I see, then the sun that I see is not the
astronomer’s sun. For exactly the same reasons, the tables and
chairs that I see, if they are identical with my seeing of them, are
not located where physics says they are, but where my seeing is.
You may say that my seeing, being mental, is not in space; if
you do, I will not argue the point. But I shall none the less insist
that there is one, and only one, region of space-time with which
my seeing is always causally bound up, and that is my brain at
the time of the seeing. And exactly the same is true of all objects
of sense-perception.

We are now in a position to consider the relation between a
physical occurrence and the subsequent occurrence popularly
regarded as seeing it. Consider, say, a flash of lightning on a
dark night. The flash, for the physicist, is an electrical discharge,
which causes electromagnetic waves to travel outward from the
region where it has taken place. These waves, if they meet no
opaque matter, merely travel further and further; but when they
meet opaque matter their energy undergoes transformations into
new forms. When they happen to meet a human eye connected
with a human brain, all sorts of complicated things happen, which
can be studied by the physiologist. At the moment when this
causal process reaches the brain, the person to whom the brain
belongs “sees” the flash. This person, if he is unacquainted with
physics, thinks that the flash is what takes place when he “sees”
the flash ; or rather, he thinks that what takes place is a relation
between himself and the flash, called “perceiving” the flash. If
he is acquainted with physics, he does not think this, but he still
holds that the sort of thing that takes place when he “sees” the
flash gives an adequate basis for knowledge of the physical world.

We can now at last tackle the question: How, and to what
extent, can percepts be a source of knowledge as to physical
objects? A percept, we have agreed, comes at the end of a causal
chain which starts at the object. (Of course no causal chain really
has either a beginning or an end. From another point of view the
percept is a beginning; it begins the reaction to a stimulus.) If
the percept is to be a source of knowledge of the object, it must

221

HUMAN knowledge: its scope and limits

be possible to infer the cause from the effect, or at least to infer
some characteristics of the cause. In this backward inference
from effect to cause, I shall for the present assume the laws of
physics.

If percepts are to allow inferences to objects, the physical
world must contain more or less separable causal chains. I can
see at the present moment various things — sheets of paper, books,
trees, walls, and clouds. If the separateness of these things in my
visual field is to correspond to a physical separateness, each of
them must start its own causal chain, arriving at my eye without
much interference from the others. The theory of light assures
us that this is the case. Light- waves emanating from a source will,
in suitable circumstances, pursue their course practically un-
affected by other light-waves in the same region. But when light-
waves encounter a reflecting or refracting object this independence
of the medium disappears.

This is important in deciding what the object is that we are
supposed to see. In the daytime, practically all the light that
reaches the eye comes ultimately from the sun, but we do not say
that we are seeing only the sun. We are seeing the last region
after which the course of the light was virtually unimpeded until
it reached the eye. When light is reflected or scattered, we con-
sider, as a rule, that it makes us see the last object from which it
is reflected or scattered ; when it is refracted, we consider that we
are still seeing the previous source, though inaccurately. Re-
flected light, however, is not always taken as giving perception
of the reflector; it is not so taken when the reflection is accurate,
as in a mirror. What I see when I shave I consider to be my own
face. But when sunlight is reflected on an outdoor landscape it
gives me much more information about the things in the landscape
than about the sun, and I therefore consider that I am perceiving
the things in the landscape.

In a lesser degree similar things may be said about sound. We
distinguish between hearing a sound and hearing an echo of it.
If the sun were as chromatically noisy as it is bright, and if terres-
trial things were resonant only to certain of its notes, we should
say that we were hearing the things, not the sun, when they gave
characteristic sound-reflections.

The other senses do not give the same kind of perception of
distant objects or of intermediate links in causal chains, because

222

PHYSICS AND EXPERIENCE

they are not concerned with physical processes having the peculiar
kind of independence that is characteristic of wave motions.

From what we have been saying it is clear that the relation of
a percept to the physical object which is supposed to be perceived
is vague, approximate, and somewhat indefinite. There is no
precise sense in which we can be said to perceive physical objects.

The question of perception as a source of knowledge can be
merged in a wider question : How far, and in what circumstances,
can one stage in a physical process be a basis for inferring an
earlier stage ? Clearly this can only happen in so far as the process
in question is independent of other processes. That processes can
be thus independent is perhaps surprising. We see the separate
stars because the light that starts from each travels on through
regions full of other trails of light, and yet retains its independence.
When this independence fails, we see a vague blur, like the milky
way. In the case of the milky way, the independence does not fail
till we reach the physiological stage; that is why telescopes can
separate the different stars of the milky way. But the independence
of the light from different parts of one star cannot be restored by
telescopes ; that is why stars have no measurable apparent magni-
tude.

Our perceptive apparatus, as studied by the physiologist, can
to some extent be ignored by the physicist, because it can be
treated as approximately constant. It is not of course really
constant. By squinting I can see two suns, but I do not imagine
that I have performed an astronomical miracle. If I close my eyes
and turn my face to the sun, I see a vague red glare; this change
in the sun's appearance I attribute to myself, not to the sun.
Things look different when I see them out of the corner of my eye
from what they do when I focus on them. They look different to
short-sighted and to long-sighted people. And so on. But common
sense learns to distinguish these subjective sources of variation
in the percept from those that are due to variation in the physical
object. Until we learn to draw, we think that a rectangular object
always looks rectangular; and we are right, in the sense that an
animal inference causes us to judge it to be rectangular.

Science deals with these matters by assuming a normal observer
who is to some extent a fiction, like the economic man, but not so
completely a fiction as to be practically useless. When a normal
observer sees a difference between two objects, for example that

223

human knowledge: its scope and limits

one looks yellow and the other looks blue, this difference is assumed
to have its source in a difference in the objects, not in the subjective
perceptive apparatus of the observer. If, in a given case, this
assumption is erroneous, it is held that multiplicity of observations
by a multitude of observers will correct it. By such methods, the
physicist is enabled to treat our perceptive apparatus as the source
of a constant error, which, because it is constant, is for many
purposes negligible.

The principles which justify the inference from percepts to
physical objects have not been sufficiently studied. Why, for
example, when a number of people see the sun, should we believe
that there is a sun outside their percepts, and not merely that
there are laws determining the circumstances in which we shall
have the experience called “seeing the sun”?

Here we come up against a principle which is used both by
science and by common sense, to the effect that, when a number
of phenomena in separated parts of space-time are obviously
causally interconnected, there must be some continuous process
in the intervening regions which links them all together. This
principle of spatio-temporal continuity needs to be re-examined
after we have considered the inference from perceptual to physical
space. In the meantime, it can be accepted as at least a first step
towards formalizing inference from perceptual to physical objects

I will conclude with a summary of the present Chapter.

Our main question was : If physics is true, how can it be known,
and what, besides physics, must we know to infer physics? This
problem arises through the physical causation of perception,
which makes it probable that physical objects differ greatly from
percepts; but if so, how can we infer physical objects from per-
cepts? Moreover, since perceiving is considered to be “mental”
while its causes are “physical”, we are confronted with the old
problem of the relation between mind and matter. My own belief
is that the “mental” and the “physical” are not so disparate as is
generally thought. I should define a “mental” occurrence as one
which some one knows otherwise than by inference ; the distinction
between “mental” and “physical” therefore belongs to theory of
knowledge, not to metaphysics.

One of the difficulties which have led to confusion was failure
to distinguish between perceptual and physical space. Perceptual
space consists of perceptible relations between parts of percepts,

224

PHYSICS AND EXPERIENCE

whereas physical space consists of inferred relations between
inferred physical things. What I see may be outside my percept
of my body, but not outside my body as a physical thing.

Percepts, considered causally, are between events in afferent
nerves (stimulus) and events in efferent nerves (reaction); their
location in causal chains is the same as that of certain events in
the brain. Percepts as a source of knowledge of physical objects
can only serve their purpose in so far as there are separable, more
or less independent, causal chains in the physical world. This
only happens approximately, and therefore the inference from
percepts to physical objects cannot be precise. Science consists
largely of devices for overcoming this initial lack of precision on
the assumption that perception gives a first approximation to
the truth.

225

Chapter V

TIME IN EXPERIENCE

T he purpose of this Chapter is to consider those features
of crude experience which form the raw material of the
concept of time, which has to go through a long elaboration
before it is fit to appear in physics or history. There are two
sources of our belief in time ; the first is the perception of change
within one specious present, the other is memory. When you
look at your watch, you can see the second-hand moving, but
only memory tells you that the minute-hand and hour-hand
have moved. Shakespeare’s timepieces had no second-hand, as
appears from the lines :

Ah! yet doth beauty, like a dial hand,

Steal from his figure, and no pace perceiv’d.

“Pace perceiv’d” is only possible when the motion is so rapid
that, though the beginning and end are noticeably different,
the lapse of time is so short that both are parts of one sensation.
No sensation, not even that caused by a flash of lightning, is
strictly instantaneous. Physiological disturbances die down
gradually, and the length of time during which we see a flash
of lightning is much greater than the length of time occupied
by the physical phenomenon.

The relation of “preceding”, or of “earlier-and-later”, is an
element in the experience of perceiving a change, and also in
the experience of remembering. Strictly speaking, we ought
also to add immediate expectation, but this is of less importance.
When I see a rapid movement, such as that of a falling star, or
of cloud-shadows in a landscape, I am aware that one part of
the movement is earlier than another, in spite of the whole being
comprised within one specious present; if I were not aware of
this, I should not know whether the movement had been from
A to B or from B to A, or even that change had occurred. When
a movement is sufficiently rapid we do not perceive change:
if you spin a penny very efficiently, it takes on the appearance
of a diaphanous sphere. If a motion is to be perceived, it must
be neither too fast nor too slow. If it satisfies this condition,

226

TIME IN EXPERIENCE

it provides experiences from which it is possible to obtain
ostensive definitions of the words for temporal relations: “preced-
ing”, “succeeding”, “before”, “after”, “earlier”, “later”. When
these words have come to be understood, we can understand
such sentences as “A precedes B” even when A and B are not
part of one specious present, provided we know what is meant
by “A” and what by “B”.

But one specious present is a very small part of one man’s life,
and for longer periods of time within our own experience we
rely upon memory. In fact, of course, we rely upon a great deal
besides memory. In regard to past engagements noted in my
diary, I can infer their time-order and their distance from the
present by the dates at which they are recorded. This, however,
is a process presupposing considerable knowledge, whereas I
am concerned at present with the data upon which our knowledge
concerning time is based. Within limits, and with a considerable
risk of error, we can place our memories in a time-order by the
way they “feel”. Suppose we have just had a conversation,
beginning amicably, but ending in a violent quarrel, and suppose
the person with whom we were conversing has flounced out of
the room in a rage. We can go over the whole conversation in
retrospect, thinking “at this point I said the wrong thing”, or
“at that point the remark he allowed himself was unpardonable”.
Our memory, in fact, is not of a heap of events, but of a series ,
and often there can be no reasonable doubt that the time-order
supplied by our memory is correct.

There is here, however, a complication which has been too
often overlooked. All my recollections occur nbw 9 not at the
times when the recollected events occurred.^The time-order of
the past events, in so far as I can know it by means of memory,
must be connected with a quality of my recollections: some
must feel recent and others must feel remote. It must be by
means of this felt quality of recentness or remoteness that I
place remembered events in a series when I am relying upon
memory alone. In travelling from percepts towards “the dark
backward and abysm of time”, the present contents of my mind
have an order, which I believe to be correlated, roughly at any
rate, with the objective time-order of the events to which my
recollections refer. This order in the present contents of my
mind, which, by means of expectation, may be extended into

227

HUMAN knowledge: its scope and limits

the future, may be called “subjective” time. Its relations to
objective time are difficult, and demand discussion.

St. Augustine, whose absorption in the sense of sin led him
to excessive subjectivity, was content to substitute subjective
time for the time of history and physics. Memory, perception,
and expectation, according to him, made up all that there is of
time. But obviously this won’t do. All his memories and all his
expectations occurred at about the time of the fall of Rome,
whereas mine occur at about the time of the fall of industrial
civilization, which formed no part of the Bishop of Hippo’s
expectations. Subjective time might suffice for a solipsist of the
moment, but not for a man who believes in a real past and future,
even if only his own. My momentary experience contains a
space of perception, which is not the space of physics, and a
time of perception and recollection, which is not the time of
physics and history. My past, as it occurred, cannot be identified
with my recollections of it, and my objective history, which was
in objective time, differs from the subjective history of my present
recollections, which, objectively, is all now.

That memory is in the main veridical is, in my opinion, one
of the premisses of knowledge. What this premiss asserts is,
or implies, broadly speaking, that a present recollection is as a
rule correlated with a past event. Obviously this is not logically
necessary. I might have come into existence a few moments ago,
complete with just those recollections which I then had. If the
whole world came into existence then, just as it then was, there
will never be anything to prove that it did not exist earlier; in
fact, all the evidence that we now have in favour of its having
existed earlier, we should then have. This illustrates what I mean
by saying that memory is a premiss, for we are none of us
prepared for a moment to entertain the supposition that the
world began five minutes ago. We do not entertain the supposi-
tion, because we are persuaded that, as a rule, when we recollect,
something resembling our present recollection occurred at a
time which is objectively past.

I said a moment ago that the general trustworthiness of memory
is a premiss of human knowledge. We might find at a later stage
that it can be subsumed under a wider premiss, but for the present
this possibility may be ignored. What does, however, need to
be considered at this stage is the relation of confidence in

228

TIME IN EXPERIENCE

particular memories to the postulate that memory is trustworthy
as a rule, or in certain specified circumstances.

When I remember something, I do not first take note of my
present state of mind, then reflect that memory is usually
veridical, and finally infer that something like what I am re-
collecting occurred in the past. On the contrary, what happens
when I remember is belief that something happened in the past.
What I am concerned with in this Chapter is (a) analysis of such
beliefs, and (b) statement of what is meant when such a belief is
said to be true. Neither of these questions is as simple as seems
to be generally supposed.

Memories often float through the mind as mere images, un-
accompanied by belief, but I am concerned only with memories
that are believed. Let us take a concrete illustration. Suppose I
have seen my child very nearly run over by a motor-car, but in
fact unhurt, and suppose that in the following night I have a
nightmare in which the child is killed. When I wake up, I think,
with inexpressible relief: “ This did not occur; that occurred.”

A good deal of clarification is necessary before we can arrive
at the core of the problems raised by this illustration. To begin
with, when we say “this did not occur”, we are not denying that
the nightmare occurred; in so far as we are remembering the
nightmare as a private experience, our memory is quite correct.
The nightmare, however, did not have the context that waking
experiences have: it had no context whatever in the life of the
child, or of any person except myself and such persons as had to
listen while I related it, and in my own life its context sharply
ended when I woke, instead of being prolonged through years
of sorrow. This is the sort of thing we mean by saying that the
nightmare was only a dream.

But all this is irrelevant to our problem of memory, and I
have only mentioned it in order to make clear what is relevant
and what is not. When I remember the nightmare, my memory
is veridical ; I am only misled if I suppose the nightmare to have
had the sort of context that a similar waking experience would
have had. An error of memory occurs only when we believe that,
in the past, we had some experience which in fact we did not
have, and when, further, we believe this in the specific way that
is called “remembering”, as opposed to the way that comes when
we read records of forgotten events, or have to listen to aunts

229

human knowledge: its scope and limits

relating our exploits as children. Such errors of memory un-
doubtedly occur. I will not insist upon George IV remembering
that he was at the battle of Waterloo; coming nearer home, I
know that when, too late, I have thought of a witty retort, I
find a tendency to remember that I actually made it, which I
only resist by a considerable moral effort. When two people
independently report an acrimonious conversation, each will
usually falsify the truth in a way favourable to his self-esteem.
And even memories having little emotional interest can often
be proved by records to be inaccurate.

But the most convincing examples of false memories are
supplied by dreams, though not by the nightmare which I
supposed a moment ago. Let us alter the nightmare: I now do
not dream that I see the child killed, but that, having seen this,
it is my duty to tell the child’s mother what has happened. This
is just as much a nightmare, but now the false belief in my dream
is not merely as to the context of my experience, but as to my
very own past experience. When I dream that I see the child
run over, I do have the experience, though it does not have its
usual concomitants; but when I dream that I saw the child
run over, I never did have the experience that in my dream I
am remembering. This is a genuine case of false memory, and
shows that memory alone cannot make it certain that what is
remembered really occurred, however much we may whittle down
what is remembered to its core of purely personal experience.

This example, I hope, will make it clear what I mean by “sub-
jective” time, and what is the problem of its relation to objective
time. In dreams, as in waking life, there is a difference between
perceiving and remembering. The perceiving and the remembering
do really occur in dreams, and so far as the perceiving is concerned
we do not have to suppose that dreams deceive us as to our own
experiences: what, in dreams, we see and hear, we do in fact
see and hear, though, owing to the unusual context, what we see
and hear gives rise to false beliefs. Similarly what we remember
in dreams we do really remember, that is to say, the experience
called “remembering” does occur. In the dream, this remembering
has a quality differing from that of dream-perception, and in
virtue of this quality the remembering is referred to the past.
But the quality is not that of genuine pastness which belongs to
the events of history; it is that of subjective pastness, in virtue

230

TIME IN EXPERIENCE

of which the present remembering is judged (falsely) to refer to
something that is objectively past.

This quality of subjective pastness belongs to waking memories
as well as to those of dreams, and is what makes them subjectively
distinguishable from perceptions. It is a quality which is capable
of degrees: our memories feel more remote or less remote, and
can be arranged in a series by this qualitative difference. But since
all our memories are now from the point of view of history, this
subjective time-order is wholly distinct from the objective time-
order, though we hope that there is a certain degree of correspond-
ence between the two.

I can perceive a remembering, but I cannot perceive what is
remembered. Remembering consists of remembering “some-
thing”. I want now to analyse remembering, and especially to
consider this relation to “something”. In short: what do we
mean when, thinking of some past occurrence, we judge “ that
occurred”? What can “that” be?

The difficulty is that, to know what we mean by “that occurred”,
the word “that” must refer to some present content of the mind,
whereas, if the word “occurred” is justified, the word “that” must
refer to something in the past. Thus it would seem that the word
“that” must refer to something which is both past and present.
But we are in the habit of thinking that the past is dead, and that
nothing past can also be present. What, then, do we mean when
we judge “that occurred”?

There are two different possible answers, connected with the
two different theories as to proper names which we considered
in an earlier chapter. If we consider that, in describing the struc-
ture of the world, the terms which must be merely named must
include “events”, which are uniquely defined by their spatio-
temporal position, and are logically incapable of recurring, then
we must say that the phrase “that occurred” is inaccurate, and
should be replaced by “something very like that occurred”.
If, on the other hand, we hold that an “event” can be defined
as a bundle of qualities, each and all of which may recur, then
“that occurred” may be completely accurate. If, for example,
I see a rainbow on two occasions, and I see some shade of colour
near the middle on one occasion, I probably saw the very same
shade on the other occasion. If, then, remembering the earlier
rainbow while I am seeing the later one, I say, of some shade of

231

human knowledge: its scope and limits
*

colour that I am now seeing, “that occurred on the previous
occasion”, what I say may be exactly true.

Either of these answers will solve the particular difficulty
with which we have been immediately concerned, and for the
present I shall not attempt to decide between them. They leave
open the question what is meant by the word “occurred”; I
shall deal with this when I come to consider public time.

It should be observed that what we mean by “the past” in the
historical sense is understood by us in virtue of the experience
of succession within one specious present. It is this experience
that makes us understand the word “precedes”. We can then
understand: “if y is in the specious present, x precedes y”. We
can therefore understand what is meant by saying that x precedes
everything in the specious present, i.e. that x is in the past.
The essential point is that the time that occurs in the specious
present is objective, not subjective.

We can now sum up the discussion of this Chapter.

There are two sources for our knowledge of time. One is the
perception of succession within one specious present, the other
is memory. Remembering can be perceived, and is found to have
a quality of greater or less remoteness, in virtue of which all my
present memories can be placed in a time-order. But this time is
subjective, and must be distinguished from historical time.
Historical time has to the present the relation of “preceding”,
which I know from the experience of change within one specious
present. In historical time, all my present memories are now ,
but in so far as they are veridical they point to occurrences in
the historical past. There is no logical reason why any memories
should be veridical; so far as logic can show, all my present
memories might be just what they are if there had never been
any historical past. Our knowledge of the past therefore depends
upon some postulate which is not to be discovered by mere
analysis of our present rememberings.

23*

Chapter VI

SPACE IN PSYCHOLOGY

P sychology is concerned with space, not as a system of
relations among material objects, but as a feature of our
perceptions. If we could accept naive realism, this dis-
tinction would have little importance : we should perceive material
objects and their spatial relations, and the space that characterizes
our perceptions would be identical with the space of physics.
But in fact naive realism cannot be accepted, percepts are not
identical with material objects, and the relation of perceptual
to physical space is not identity. What the relation is, I shall
consider presently; I am concerned, to begin with, only with
space as it appears in psychology, ignoring all questions of
physics.

It is clear that experience is what has led us to believe in the
existence of spatial relations. Psychology is concerned to examine
what experiences are relevant, and by what process of inference
or construction we pass from such experiences to the space of
common sense. Since a great part of the process occurs in early
infancy, and is no longer remembered in later years, it is a some-
what difficult matter of observation and inference to discover the
character of the original experiences which give rise to the habits
that adult common sense takes for granted. To take only the most
obvious instance : we place things touched and things seen in one
space, automatically and without reflection, but babies under the
age of about three months seem unable to do so. That is to say,
they do not know how to touch an object that they see and that
is within their reach. It is only through frequent accidental
contacts that they gradually learn the movements necessary to
produce a tactile sensation when a visual sensation is given.
Chickens, on the other hand, can do this from birth.

We have to separate the crude material of sensation from the
supplementation that it has acquired through experience and
habit. When you see (let us say) an orange, you do not have
merely a visual experience, but also expectations of touch, smell,
and taste. You would have a violent shock of surprise if you
found that it felt like putty, or smelt like a bad egg, or tasted

233

HUMAN knowledge: its scope and limits

like a beef-steak. You would be even more surprised if, like
Macbeth’s dagger, it proved incapable of being touched. Such
surprises show that expectations of non-visual sensations are
part of what spontaneously happens to you when you have a
visual sensation of a familiar kind. In the chicken, apparently,
such expectations are in part due to its innate constitution. In
human beings this happens much less, if at all; our expectations
seem to be generated, mainly if not wholly, by experience. A
visual sensation has at first, it would seem, a certain purity, and
only gradually, through frequent collocations, acquires the
penumbra of expectations connected with other senses that it
has in adult life. And the same is true of other senses.

It follows that the unitary space of common sense is a con-
struction, though not a deliberate one. It is part of the business
of psychology to make us aware of the steps in this construction.

When we examine our momentary visual field, stripping it,
as far as we can, of all the adjuncts derived from experience, we
find that it is a complex whole in which the parts are inter-
related in various ways. There are relations of right and left,
up and down; there are also relations which we learn to interpret
as depth. These relations all belong to the sensational datum.
The best way to become aware of the sensational element in
visual perceptions of depth is by the use of the stereoscope.
When you look at the two separate photographs that are going
to be seen together in the stereoscope, they both look flat, as
they are; but when you see the combination in the stereoscope,
you get the impression that things “stand out”, and that some
are nearer than others. As a judgment this would of course be
mistaken; the photographs are just as flat as they were before.
But it is a genuine quality of the visual datum, and very instruc-
tive as a help in showing how we arrive visually at estimates of
depth.

By means of the three relations right-and-left, up-and-down,
seeming-far-and-seeming-near, your momentary visual field can
be arranged in a three-dimensional manifold. But far-and-near,
estimated visually, is not capable of distinguishing except when
one of the distances is very short; we cannot “see” that the sun
is further off than the moon, or even than clouds which are not
obscuring it.

Other senses than sight supply other elements which contri-

*34

SPACE IN PSYCHOLOGY

bute to the common-sense construction of space. When a part of
the body is touched, we can tell, within limits, what part it is,
without needing to look. (On the tongue or the finger-tips we
can tell pretty accurately, on the back only vaguely.) This implies
that touches in one part have a quality not belonging to touches
in another part, and that the qualities appropriate to different
parts have relations enabling us to arrange them in a two-dimen-
sional order. Experience teaches us to connect sensations of touch
with the visual sensations of seeing different parts of the body.

Not only static sensations, such as we have mentioned, but
also sensations of movement are involved in constructing
common-sense space. Sensations of movement are of two sorts,
active and passive — active when we have a feeling of muscular
exertion, passive when the observed change seems independent
of ourselves. When we move a part of our body and see it move,
we have active and passive sensations at the same time. What
I am calling passive sensations are only relatively passive; there
is always the activity of attention, involving adjustment of the
sense organs, except in the case of rather violent sensations.
When you unexpectedly hit your head against a low doorway
you are almost wholly passive, whereas when you listen carefully
to a very faint sound the element of activity is considerable.
(I am speaking of activity and passivity as elements in sensations,
and am not inquiring into their causal status.)

Movement is essential in enlarging our conception of space
beyond our own immediate neighbourhood. The distance from
where we are to some place may be estimated as an hour’s
walk, three hours by train, or twelve hours by aeroplane. All
such estimates assume fixed places. You can tell how long it
takes from London to Edinburgh, because both retain fixed
positions on the earth’s surface, but you cannot tell how long it
will take to reach Mr. Jones, because he may move while you
are on the way. All distances above a rather small minimum
depend upon the assumption of immobility; it is partly the
fact that this assumption is never quite true that necessitated
the special theory of relativity, in which distance is between
events, not bodies, and is a space-time distance, not a purely
spatial distance. But such considerations take us beyond the
scope of common sense.

It is to be observed that the spatial relations given in sensation

*35

HUMAN knowledge: its scope and limits

are always between data of the same sense. There is a spatial
relation between two parts of the same visual field, or between
two simultaneous pin-pricks on different parts of the hand;
such spatial relations are within the realm of sensation, and are
not learnt by experience. But between the tactual sensation of
a pin-prick and the visual sensation of seeing the pin there is
no direct sensational spatial relation, but only a correlation
which human beings learn by experience. When you both see and
feel a pin touching your hand it is only experience that enables
you to identify the point of contact seen and the point of contact
felt by touch. To say that they are the same place is convenient,
but in psychology it is not strictly accurate: what is accurate
is that they are correlated places in two different spaces, visual
and tactual. It is true that in physical space only one place is
involved, but this place lies outside our direct experience, and
is neither visual nor tactual.

The construction of one space in which all our perceptual
experiences are located is a triumph of pre-scientific common
sense. Its merit lies in its convenience, not in any ultimate truth
that it may be supposed to possess. Common sense, in attributing
to it a degree of non-conventional truth beyond what it actually
has a right to claim, is in error, and this error, uncorrected, adds
greatly to the difficulty of a sound philosophy of space.

An even more serious error, committed not only by common
sense but by many philosophers, consists in supposing that the
space in which perceptual experiences are located can be identified
with the inferred space of physics, which is inhabited mainly
by things which cannot be perceived. The coloured surface
that I see when I look at a table has a spatial position in the
space of my visual field ; it exists only where eyes and nerves and
brain exist to cause the energy of photons to undergo certain
transformations. (The “where” in this sentence is a “where”
in physical space.) The table as a physical object, consisting of
electrons, positrons, and neutrons, lies outside my experience,
and if there is a space which contains both it and my perceptual
space, then in that space the physical table must be wholly
external to my perceptual space. This conclusion is inevitable
if we accept the view as to the physical causation of sensations
which is forced on us by physiology and which we considered
in an earlier chapter.

236

SPACE IN PSYCHOLOGY

The conception of one unitary space, Kant’s “infinite given
whole”, is one which must be abandoned. The crude material
available for empirical constructions contains several kinds of
relations — more especially those between parts of one visual
field or parts of one tactual field — each of which arranges its
field in a manifold having the properties that pure mathematicians
need for a geometry. By means of correlations — more especially
between the visual and tactual place of an object which I simultan-
eously see and touch — the various spaces generated by relations
of parts of single sensational fields can be amalgamated into one
space. To the making of this space experience of correlations is
necessary; the kinds of relations given in single experiences no
longer suffice.

The common- sense world results from a further correlation,
combined with an illegitimate identification. There is a correlation
between the spatial relations of unperceived physical objects
and the spatial relations of visual or other sensational data, and
there is an identification of such data with certain physical objects.
For example : I am sitting in a room, and I see — or at least common
sense thinks I see — spatial relations between the pieces of furniture
that it contains. I know that on the other side of the door there
is a hall and a staircase. I believe that the spatial relations of
things beyond the door — e.g. the relation “to-the-left-of” —
are the same as those between the bits of furniture that I see;
and further, I identify what I see with physical objects which
can exist unseen, so that, if I am content with common sense,
there is no gulf between the visual furniture and the unseen
hall beyond the door. The two accordingly are thought to fit
into one space, of which part is perceived while the rest is
inferred.

But in fact, if physics and physiology are to be believed, I
do not “see” the furniture in my room except in a Pickwickian
sense. When I am said to “see” a table, what really happens is
that I have a complex sensation which is, in certain respects,
similar in structure to the physical table. The physical table,
consisting of electrons, positrons, and neutrons, is inferred,
and so is the space in which it is located. It has long been a
commonplace in philosophy that the physical table does not
have the qualities of the sensational table: it has no colour, it
is not warm or cold in the sense in which we know warmth

237

HUMAN knowledge: its scope and limits

and cold by experience, it is not hard or soft if “hard” and “soft”
mean qualities given in tactile sensations, and so on. All this,
I say, has long been a commonplace, but it has a consequence
that has not been adequately recognized, namely that the space
in which the physical table is located must also be different
from the space that we know by experience.

We say that the table is “outside” me, in a sense in which my
own body is not. But in saying this we have to guard against an
ambiguity due to the necessity of distinguishing between physical
and psychological space. The visual table is “outside” my body
in visual space, if “my body” is interpreted as what I see, and
not as what physics takes to be my body. The physical table
is “outside” my body if my body is interpreted as in physics,
but has no direct or obvious spatial relation to my body as a visual
object that I experience. When we come to consider the hall on
the other side of the door, which I cannot see, we are wholly
confined to the physical sense: the hall is outside my physical
body in physical space, but is not, in any obvious sense, outside
my sensational body in psychological space, because there is no
sensational hall, and therefore the hall has no location whatever
in psychological space. Thus while there are two senses in which
the table is “outside” me, there is only one sense in which the
hall is “outside” me.

There is a further source of confusion, which is due to the fact
that there are two quite divergent ways of correlating psychological
and physical space. The obvious way is to correlate the place of
the sensational table in psychological space with the place of the
physical table in physical space, and for most purposes this is
the more important correlation. But there is a quite different
relation between the two kinds of space, and this other relation
must be understood if confusions are to be avoided. Physical
space is wholly inferential, and is constructed by means of causal
laws. Physics starts with a manifold of events, some of which
can be collected into series by physical laws; for example, the
successive events constituting the arrival of a light-ray at suc-
cessive places are bound together by the laws of the propagation
of light. In such cases we use the denial of action at a distance,
not as a physical principle, but as a means of defining space-time
order. That is to say, if two events are connected by a causal
law, so that one is an effect of the other, any third event which

238

SPACE IN PSYCHOLOGY

is a cause of the one and an effect of the other is to be placed
between the two in space-time order.

Consider now a single causal sequence, beginning with an
external stimulus, say to the eye, continuing along afferent nerves
to the brain, producing first a sensation and then a volition,
followed by a current along efferent nerves and finally a muscular
movement. This whole series, considered as one causal sequence,
must, in physical space-time, occupy a continuous series of
positions, and since the physiological terms of the series end and
begin in the brain, the “mental” terms must begin and end in
the brain. That is to say, considered as part of the manifold of
events ordered in space-time by causal relations, sensations and
volitions must be located in the brain. A point in space-time,
following the theory to be developed in a subsequent chapter,
is a class of events, and there is no reason why some of these
events should not be “mental”. Our feeling to the contrary is
only due to obstinate adherence to the mind-matter dualism.

We can now sum up the above discussion. When I have the
experience called “seeing a table”, the visual table has, primarily,
a position in the space of my momentary visual field. Then,
by means of experienced correlations, it has a position in a space
which embraces all my perceptual experiences. Next, by means
of physical laws it is correlated with a place in physical space-
time, namely the place occupied by the physical table. Finally,
by means of psychophysical laws it is related to another place
in physical space-time, namely the place occupied by my brain
as a physical object. If the philosophy of space is to avoid hopeless
confusions, these different correlations must be kept carefully
disentangled.

In conclusion, it should be observed that the twofold space
in which percepts are located is closely analogous to the twofold
time of memories. In subjective time, memories are in the past;
in objective time, they are now. Similarly, in subjective space
my percept of a table is over there, but in physical space it is here.

239

Chapter VII

MIND AND MATTER

C ommon sense believes that we know something about
mind, and something about matter; it holds, further,
that what we know of both is enough to show that they
are quite different kinds of things. I hold, on the contrary, that
whatever we know without inference is mental, and that the
physical world is only known as regards certain abstract features
of its space-time structure — features which, because of their
abstractness, do not suffice to show whether the physical world
is, or is not, different in intrinsic character from the world of
mind.

I will begin with an attempt to state the common-sense point
of view as clearly as is possible in view of the confusions that
are essential to it.

Mind — so common sense might say — is exhibited by persons
who do and suffer various things. Cognitively, they perceive,
remember, imagine, abstract, and infer; on the side of the
emotions, they have feelings that are pleasurable and feelings
that are painful, and they have sentiments, passions, and desires;
volitionally, they can will to do something or will to abstain
from doing something. All these occurrences can be perceived
by the person to whom they happen, and all are to be classified
together as “mental” events. Every mental event happens “to”
some person, and is an event in his life.

But in addition to perceiving “thoughts” — so common sense
holds — we also perceive “things”, and events which are outside
ourselves. We see and touch physical objects; we hear sounds
which are also heard by other people, and therefore are not
in us; when we smell a bad drain, other people do so too, unless
they are plumbers. What we perceive, when it is outside our-
selves, is called “physical”; this term includes both “things”
which are “matter”, and events, such as a noise or a flash of
lightning.

Common sense also allows inferences to what is not perceived,
at any rate by us, e.g. the centre of the earth, the other side of
the moon, the thoughts of our friends, and the mental events

240

MIND AND MATTER

that have produced historical records. An inferred mental event
can be known without inference by the person to whom it happens.
An inferred physical thing or event may or may not have been
perceived by some one; some physical things, such as the centre
of the earth, are held to have been never perceived.

This common-sense view, while on the whole acceptable
as regards mental events, requires radical alteration where
physical events are concerned. What I know without inference
when I have the experience called “seeing the sun” is not the
sun, but a mental event in me. I am not immediately aware of
tables and chairs, but only of certain effects that they have on me.
The objects of perception which I take to be “external” to me,
such as coloured surfaces that I see, are only “external” in my
private space, which ceases to exist when I die — indeed my
private visual space ceases to exist whenever I am in the dark
or shut my eyes. And they are not “external” to “me”, if “me”
means the sum-total of my mental events; on the contrary, they
are among the mental events that constitute me. They are only
“external” to certain other percepts of mine, namely those which
common sense regards as percepts of my body; and even to
these they are “external” only for psychology, not for physics,
since the space in which they are located is the private space of
psychology.

In considering what common sense regards as perception of
external objects, there are two opposite questions to be considered :
first, why must the datum be regarded as private; second, what
reason is there to take the datum as a sign of something which
has an existence not dependent upon me and my perceptive
apparatus ?

The reasons for regarding the datum — say in sight or touch —
as private, are twofold. On the one hand there is physics, which,
starting with the intention of doing its best for naive realism,
arrives at a theory of what goes on in the physical world which
shows that there is no ground for supposing the physical table
or chair to resemble the percept except in certain abstract
structural respects. On the other hand there is the comparison
of what different people experience when, according to common
sense, they perceive the same thing. If we confine ourselves to
the sense of sight, when two people are said to see the same
table there are differences of perspective, differences of apparent

241 Q

HUMAN knowledge: its scope and limits

size, differences in th way light is reflected, and so on. Thus
at most the projective properties of the table are the same for a
number of percipients, and even these are not quite the same
if there is a refracting medium such as a steaming kettle, or our
old friend the water that makes a stick look bent. If we consider,
as common sense does, that the “same” object can be perceived
by both sight and touch, the object, if it is to be really the same,
must be still further removed from the datum, for a complex
sight-datum and a complex touch-datum differ in intrinsic
quality, and cannot be similar except in structure.

Our second question is more difficult. If the datum in my
perceptions is always private to me, why do I nevertheless regard
it as a sign by means of which I can infer a physical “thing” or
event which is a cause of my percept if my body is suitably
placed, but does not form part of my immediate experience
except partially in exceptional cases ?

When we begin to reflect, we find ourselves with an unshake-
able conviction that some of our sensations have causes external
to our own body. Headache, toothache, and stomachache, we
are willing to admit, have internal causes, but when we stub our
toe or run into a post in the dark or see a flash of lightning we
cannot easily make ourselves doubt that our experience has an
external source. It is true that we sometimes come to think
that this belief was mistaken, for instance if it occurs in dreams,
or when we have a buzzing in the ears which sounds like the
hum of telegraph wires. But such cases are exceptional, and
common sense has discovered ways of dealing with them.

What chiefly confirms us in our belief that most sensations
have physical causes is, on the one hand, the quasi-publicity
of many sensations, and, on the other hand, the fact that, if
regarded as arising spontaneously, they seem completely erratic
and unaccountable.

As regards quasi-publicity, the argument is the opposite of
that which proves the privacy of data : although two neighbouring
men do not have exactly the same visual data, they have data
which are very similar, and although visual and tactual qualities
are different, the structural properties of an object seen are
approximately identical with those of the same object touched.
If you have models of the regular solids, the one which you can
see to be a dodecahedron will be correctly named by an educated

242

MIND AND MATTER

blind man after feeling it. Apart from the publicity concerned
with different percipients, there is also what may be called
temporal publicity in one man’s experience. I know that by taking
suitable measures I can see St. Paul’s at any time; I know that
the sun and moon and stars recur in my visual world, and so
do my friends and my house and my furniture. I know that the
differences between the times when I see these objects and the
times when I do not are easily explicable as due to differences
in me or my environment which do not imply any change in
the objects. Such considerations confirm the common-sense
belief that there are, in addition to mental events, things which
are sources of similar percepts in different observers at one time,
and often of the same observer at different times.

As regards the irregularity of a world consisting only of data,
this is an argument to which it is difficult to give precision.
Roughly speaking, many sensations occur without any fixed
antecedents in our own experience, and in a manner which
suggests irresistibly that, if they have causes, these causes lie
partly outside our experience. If you are hit on the head by a
tile falling off a roof while you are walking below, you will
experience a sudden violent pain which cannot be explained
causally by anything of which you were aware before the accident
happened. It is true that there are some extreme psycho-analysts
who maintain that accidents only happen to people who have
grown tired of life through reflecting on their sins, but I do not
think such a view has many adherents. Consider the inhabitants
of Hiroshima when the bomb burst: it cannot be that they had
all reached a point in their psychological development which
demanded disaster as the next step. To explain such an occur-
rence causally, we must admit purely physical causes; if they
are rejected, we must acquiesce in causal chaos.

Such arguments may be reinforced by the considerations
set forth above in the chapter on solipsism, showing that we
must choose between two alternatives: either (a) no inferences
from data to other events are to be admitted as valid, in which
case we know far less than most solipsists suppose, and in fact
a great deal less than we can force ourselves to regard as the
minimum of our knowledge, or ( b ) there are principles of inference
which allow us to infer things outside our own experience.

Belief in the physical causation of sensations is also reinforced

243

human knowledge: its scope and limits

by the fact that, if this belief is rejected, there remains no reason
for the acceptance of science in its broad outlines, while the
refusal of such acceptance does not seem rational.

Such are the broad considerations which lead us to look for a
way of systematizing and rationalizing our common-sense pro-
pensity to infer physical causes of sensations.

The inferences from experiences to the physical world can,
I think, all be justified by the assumption that there are causal
chains, each member of which is a complex structure ordered
by the spatio-temporal relation of compresence (or of conti-
guity); that all the members of such a chain are similar in struc-
ture ; that each member is connected with each other by a series
of contiguous structures; and that, when a number of such
similar structures are found to be grouped about a centre earlier
in time than any of them, it is probable that they all have their
causal origin in a complex event which is at that centre and has
a structure similar to the structure of the observed events. I shall,
at a later stage, endeavour to give greater precision to this assump-
tion, and to show reasons for accepting it. For the present, to
avoid verbiage, I shall treat it as though it were unquestionably
correct, and on this basis I shall return to the relations between
mental and physical events.

When, on a common-sense basis, people talk of the gulf between
mind and matter, what they really have in mind is the gulf between
a visual or tactual percept and a “thought” — e.g. a memory, a
pleasure, or a volition. But this, as we have seen, is a division
within the mental world ; the percept is as mental as the “thought”.
Slightly more sophisticated people may think of matter as the
unknown cause of sensation, the “thing-in-itself” which certainly
does not have the secondary qualities and perhaps does not have
the primary qualities either. But however much they may
emphasize the unknown character of the thing-in-itself, they
still suppose themselves to know enough of it to be sure that it
is very different from a mind. This comes, I think, of not having
rid their imaginations of the conception of material things as
something hard that you can bump into. You can bump into
your friend's body, but not into his mind; therefore his body is
different from his mind. This sort of argument persists imagina-
tively in many people who have rejected it intellectually.

Then, again, there is the argument about brain and mind,

244

MIND AND MATTER

When a physiologist examines a brain, he does not see thoughts,
therefore the brain is one thing and the mind which thinks is
another. The fallacy in this argument consists in supposing
that a man can see matter. Not even the ablest physiologist
can perform this feat. His percept when he looks at a brain is
an event in his own mind, and has only a causal connection
with the brain that he fancies he is seeing. When, in a powerful
telescope, he sees a tiny luminous dot, and interprets it as a vast
nebula existing a million years ago, he realizes that what he
sees is different from what he infers. The difference from the
case of a brain looked at through a microscope is only one of
degree: there is exactly the same need of inference, by means
of the laws of physics, from the visual datum to its physical
cause. And just as no one supposes that the nebula has any close
resemblance to a luminous dot, so no one should suppose that
the brain has any close resemblance to what the physiologist
sees.

What, then, do we know about the physical world? Let us
first define more exactly what we mean by a “physical” event.
I should define it as an event which, if known to occur, is inferred,
and which is not known to be mental. And I define a “mental”
event (to repeat) as one with which some one is acquainted
otherwise than by inference. Thus a “physical” event is one
which is either totally unknown, or, if known at all, is not known
to any one except by inference — or, perhaps we should say, is
not known to be known to any one except by inference.

If physical events are to suffice as a basis for physics, and,
indeed, if we are to have any reason for believing in them, they
must not be totally unknown, like Kant’s things-in-themselves.
In fact, on the principle which we are assuming, they are known,
though perhaps incompletely, so far as their space-time structure
is concerned, for this must be similar to the space-time structure
of their effects upon percipients. E.g. from the fact that the
sun looks round in perceptual space we have a right to infer
that it is round in physical space. We have no right to make a
similar inference as regards brightness, because brightness is
not a structural property.

We cannot, however, infer that the sun is not bright— meaning
by “brightness” the quality that we know in perception. The
only legitimate inferences as regards the physical sun are

24s

HUMAN knowledge: its scope and limits

structural; concerning a property which is not structural, such
as brightness, we must remain completely agnostic. We may
perhaps say that it is unlikely that the physical sun is bright,
since we have no knowledge of the qualities of things that are
not percepts, and therefore there seems to be an illimitable field
of choice of possible qualities. But such an argument is so specula-
tive that perhaps we ought not to attach much weight to it.

This brings us to the question: Is there any reason, and if so
what, for supposing that physical events differ in quality from
mental events?

Here we must, to begin with, distinguish events in a living
brain from events elsewhere. I will begin with events in a living
brain.

I assume, for reasons which will be given in Part IV, that a
small region of space- time is a collection of compresent events,
and that space-time regions are ordered by means of causal
relations. The former assumption has the consequence that there
is no reason why thoughts should not be among the events of
which the brain consists, and the latter assumption leads to the
conclusion that, in physical space, thoughts are in the brain.
Or, more exactly, each region of the brain is a class of events,
and among the events constituting a region thoughts are included.
It is to be observed that, if we say thoughts are in the brain, we
are using an ellipsis. The correct statement is that thoughts are
among the events which, as a class, constitute a region in the
brain. A given thought, that is to say, is a member of a class,
and the class is a region in the brain. In this sense, where events
in brains are concerned, we have no reason to suppose that they
are not thoughts, but, on the contrary, have strong reason to
suppose that at least some of them are thoughts. I am using
“thoughts” as a generic term for mental events.

When we come to events in parts of physical space-time where
there are no brains, we have still no positive argument to prove
that they are not thoughts, except such as may be derived from
observation of the differences between living and dead matter
coupled with inferences based on analogy or its absence. We may
contend, for instance, that habit is in the main confined to living
matter, and that, since memory is a species of habit, it is unlikely
that there is memory except where there is living matter. Extend-
ing this argument, we can observe that the behaviour of living

246

MIND AND MATTER

matter, especially of its higher forms, is much more dependent
on its past history than that of dead matter, and that, therefore,
the whole of that large part of our mental life that depends upon
habit is presumably only to be found where there is living matter.
But such arguments are inconclusive and limited in scope. Just
as we cannot be sure that the sun is not bright, so we cannot be
sure that it is not intelligent . 1 We may be right in thinking both
improbable, but we are certainly wrong if we say they are impos-
sible.

I conclude that, while mental events and their qualities can
be known without inference, physical events are known only
as regards their space- time structure. The qualities that compose
such events are unknown — so completely unknown that we
cannot say either that they are, or that they are not, different
from the qualities that we know as belonging to mental events.

1 I do not wish the reader to take this possibility too seriously. It is
of the order of “pigs might fly”, dealt with by Mr. Crawshay-Williams
in The Comforts of Unreason , p. 193.

247

PART IV

SCIENTIFIC CONCEPTS

Chapter I

INTERPRETATION

I N all that has been said hitherto about the world of science,
everything has been taken at its face value. I am not saying
merely that we have taken the attitude of believing what men
of science tell us, for this attitude, up to a point, is the only
rational one for any man who is not a specialist on the matter in
question. In saying that this attitude is rational, I do not mean
that we should feel sure of the truth of what we are told, for it
is generally admitted that probably in due course corrections will
be found necessary. What I do mean is that the best scientific
opinion of the present time has a better chance of truth, or of
approximate truth, than any differing hypothesis suggested by
a layman. The case is analogous to that of firing at a target. If
you are a bad shot you are not likely to hit the bull's eye, but
you are nevertheless more likely to hit the bull's eye than to hit
any other equal area. So the scientist's hypothesis, though
not likely to be quite right, is more likely to be right than
any variant suggested by an unscientific person. This, how-
ever, is not the point with which we are concerned in this
chapter.

The matter with which we are now to be concerned is not
truth, but interpretation. It often happens that we have what seems
adequate reason to believe in the truth of some formula expressed
in mathematical symbols, although we are not in a position to
give a clear definition of the symbols. It happens also, in other
cases, that we can give a number of different meanings to the
symbols, all of which will make the formula true. In the former
case we lack even one definite interpretation of our formula,
whereas in the latter we have many. This situation, which may
seem odd, arises in pure mathematics and in mathematical physics;
it arises even in interpreting common-sense statements such as
“my room contains three tables and four chairs". It will thus
appear that there is a large class of statements, concerning each
of which, in some sense, we are more certain of its truth than of
its meaning. “Interpretation" is concerned with such statements;
it consists in finding as precise a meaning as possible for a state-

25 *

human knowledge: its scope and limits

ment of this sort, or, sometimes, in finding a whole system of
possible meanings.

Let us take first an illustration from pure mathematics. Mankind
have long been convinced that 2 + 2 = 4; they have been so
firmly convinced of this that it has been taken as the stock example
of something certain. But when people were asked what they
meant by “2”, “4”, “ + ”, and “ = ”, they gave vague and
divergent answers, which made it plain that they did not know
what these symbols meant. Some maintained that we know each
of the numbers by intuition, and therefore have no need to define
them. This might seem fairly plausible where small numbers
were concerned, but who could have an intuition of 3,478,921 ?
So they said we had an intuition of 1 and +; we could then
define “2” as “1 + 1”, ”3” as “2 + 1”, “4” as "3 + 1”, and
so on. But this did not work very well. It enabled us to say that
2 + 2 = (i + i) + (i + i), and that 4 = {(1 + 1) + 1} + 1
and we then needed a fresh intuition to tell us that we could
rearrange the brackets, in fact to assure us that, if /, m y n are
three numbers, then

(/ + 77?) + n = l + (m + n)

Some philosophers were able to produce this intuition on demand,
but most people remained somewhat sceptical of their claims, and
felt that some other method was called for.

A new development, more germane to our problem of inter-
pretation, was due to Peano. Peano started with three undefined
terms, “o”, “finite integer (or number)” and “successor of”, and
concerning these terms he made five assumptions, namely :

(1) o is a number;

(2) If a is a number, the successor of a (i.e. a + i) is a number;

(3) If two numbers have the same successor, the two numbers

are identical ;

(4) o is not the successor of any number;

(5) If s be a class to which belongs o and also the successor

of every number belonging to s , then every number
belongs to s.

The last of these assumptions is the principle of mathematical
induction.

252

INTERPRETATION

Peano showed that by means of these five assumptions he could
prove every formula in arithmetic.

But now a new trouble arose. It was assumed that we need not
know what we meant by “o”, “number”, and “successor”, so
long as we meant something satisfying the five assumptions. But
then it turned out that there were an infinite number of possible
interpretations. For instance, let “o” mean what we commonly
call “i”, and let “number” mean what we commonly call “number
other than o”; then all the five assumptions are still true, and all
arithmetic can be proved, though every formula will have an
unexpected meaning. “2” will mean what we usually call “3”,
but “2 + 2” will not mean “3 + 3”; it will mean “3 + 2”, and
“2 + 2 = 4” will mean what we usually express by “3 + 2 = 5”.
In like manner we could interpret arithmetic on the assumption
that “o” means “100”, and “number” means “number greater
than 99”. And so on.

So long as we remain in the region of arithmetical formulae,
all these different interpretations of “number” are equally good.
It is only when we come to the empirical uses of numbers in
enumeration that we find a reason for preferring one interpretation
to all the others. When we buy something in a shop, and the
attendant says “three shillings, please”, his “three” is not a mere
mathematical symbol, meaning “the third term after the beginning
of some series”; his “three”, in fact, is not capable of being
defined by its arithmetical properties. It is obvious that his inter-
pretation of “three” is, outside arithmetic, preferable to all the
others that Peano’s system leaves possible. Such statements as
“men have 10 fingers”, “dogs have 4 legs”, “New York has
10,000,000 inhabitants”, require a definition of numbers which
cannot be derived from the mere fact that they satisfy the formulae
of arithmetic. Such a definition is, therefore, the most satisfactory
“interpretation” of number-symbols.

The same sort of situation arises whenever mathematics is
applied to empirical material. Take, for example, geometry, con-
sidered not as a logical exercise in deducing consequences from
arbitrarily assumed axioms, but a9 a help in land surveying, map
making, engineering, or astronomy. Such practical uses of
geometry involve a difficulty which, though sometimes admitted
in a perfunctory manner, is never allowed anything like its due
weight. Geometry, as set forth by the mathematicians, uses points,

*53

human knowledge: its scope and limits

lines, planes, and circles, but it is a platitude to say that no such
objects are to be found in nature. When, in surveying, we use
the process of triangulation, it is admitted that our triangles do
not have accurate straight lines for their sides nor exact points
at their corners, but this is glozed over by saying that the sides
are approximately straight and the corners approximately points.
It is not at all clear what this means, so long as it is maintained
that there are no exact straight lines or points to which our rough'
and-ready lines and points approximate. We may mean that
sensible lines and points have approximately the properties set
forth in Euclid’s definitions and axioms, but unless we can say,
within limits, how close the approximation is, such a view will
make calculation vague and unsatisfactory.

This problem of the exactness of mathematics and the inexact-
ness of sense is an ancient one, which Plato solved by the fantastic
hypothesis of reminiscence. In modern times, like some other
unsolved problems, it has been forgotten through familiarity, like
a bad smell which you no longer notice because you have lived
with it so long. It is clear that, if geometry is to be applied to the
sensible world, we must be able to find definitions of points, lines,
planes, etc., in terms of sensible data, or else we must be able
to infer from sensible data the existence of unperceived entities
having the properties that geometry needs. To find ways, or a
way, of doing one or other of these things is the problem of the
empirical interpretation of geometry.

There is also a non-empirical interpretation, which leaves
geometry within the sphere of pure mathematics. The assemblage
of all ordered trios of real numbers forms a three-dimensional
Euclidean space. With this interpretation, all Euclidean geometry
is deducible from arithmetic. Every non-Euclidean geometry is
capable of a similar arithmetical interpretation. It can be proved
that Euclidean geometry, and every form of non-Euclidean
geometry, can be applied to every class having the same number
of terms as the real numbers; the question of the number of
dimensions, and whether the resulting geometry is Euclidean or
non-Euclidean, will depend upon the ordering relation that we
select ; an infinite number of ordering relations exist (in the logical
sense), and only reasons of empirical convenience can lead us to
select some one among them for special attention. All this is
relevant in considering what interpretation of pure geometry had

254

INTERPRETATION

better be adopted by the engineer or the physicist. It shows that,
in an empirical interpretation, the ordering relation, and not only
the terms ordered, must be defined in empirical terms.

Very similar considerations apply to time, which, however, so
far as our present question is concerned, is not so difficult a
problem as space. In mathematical physics, time is treated as
consisting of instants, though the perplexed student is assured
that instants are mathematical fictions. No attempt is made to
show him why fictions are useful, or how they are related to what
is not fictitious. He finds that by the use of these fairy tales it is
possible to calculate what really happens, and after a time he
probably ceases to trouble himself as to why this is the case.

Instants were not always regarded as fictions; Newton thought
them as “real” as the sun and moon. When this view was aban-
doned, it was easy to swing to the opposite extreme, and to forget
that a fiction which is useful is not likely to be a mere fiction.
There are degrees of fictiveness. Let us, for the moment, regard
an individual person as something in no degree Active ; what, then,
shall we say of the various aggregates of persons to which he
belongs? Most people would hesitate to regard a family as a
fictitious unit, but what about a political party or a cricket club ?
What about the assemblage of persons called “Smith”, to which
we will suppose our individual to belong? If you believe in
astrology, you will attach importance to the assemblage of persons
born under a certain planet; if you do not, you will regard such
an assemblage as Active. These distinctions are not logical; from
the logical point of view, all assemblages of individuals are equally
real or equally fictive. The importance of the distinctions is
practical, not logical : there are some assemblages about which there
are many useful things to be said, and others about which this
is not the case.

When we say that instants are useful fictions, we must be
supposed to mean that there are entities to which, as to individual
people, we feel inclined to attach a high degree of “reality”
(whatever that may mean), and that, in comparison with them,
instants have that lesser degree of “reality” that cricket clubs
have in relation to their members; but we wish also to say that
about instants, as about families as opposed to “artificial” aggre-
gates of people, there are many practically important things to say.

All this is very vague, and the problem of interpretation is that

255

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

of substituting something precise, remembering always that,
however we define “instants”, they must have the properties
required in mathematical physics. Given two interpretations which
both satisfy this requisite, the choice between them is one of taste
and convenience; there is not one interpretation which is “right”
and others that are “wrong”.

In classical physics, the technical apparatus consists of points,
instants, and particles. It is assumed that there is a three-term
relation, that of occupying a point at an instant, and what occupies
a point at an instant is called a “particle”. It is also assumed
technically that particles are indestructible, so that whatever
occupies a point at a given instant occupies some point at every
other instant. When I say that this is assumed, I do not mean
that it is asserted to be a fact, but that the technique is based on
the assumption that no harm will come of treating it as a fact.
This is still held to be the case in macroscopic physics, but in
microscopic physics “particles” have been gradually disappearing.
“Matter” in the old sense is no longer needed; what is needed
is “energy”, which is not defined except as regards its laws and
the relation of changes in its distribution to our sensations, more
especially the relation of frequencies to colour-perceptions.

Broadly speaking, we may say that the fundamental technical
apparatus of modern physics is a four-dimensional manifold of
“events” ordered by space-time relations, which can be analysed
into a spatial and a temporal component in a number of ways,
the choice between which is arbitrary. Since the calculus is still
used, it is still technically assumed that space-time is continuous,
but it is not clear how far this assumption is more than a mathe-
matical convenience. Nor is it clear that “events” have that
precise location in space-time that used to characterize a particle
at an instant. All this makes the question of the interpretation
of modern physics very difficult, but in the absence of some
interpretation we cannot say what is being asserted by the quantum
physicists.

“Interpretation”, in its logical aspect, is somewhat different
from the rather vague and difficult concept which we considered
at the beginning of this Chapter. We were there concerned with
symbolic statements which are known to have a connection with
observable phenomena, and to lead to results which observation
confirms, but are somewhat indeterminate in meaning except in

INTERPRETATION

so far as their connection with observation defines them. In this
case we can say, as we said at the beginning of this Chapter,
that we are pretty sure our formulae are true, but not at all sure
what they mean. In logic, however, we proceed differently. Our
formulae are not regarded as “true” or “false**, but as hypotheses
containing variables. A set of values of the variables which makes
the hypotheses true is an “interpretation”. The word “point”,
in geometry, may be interpreted as meaning “ordered triad of
real numbers”, or, as we shall see, as meaning what we shall call
“complete complex of compresence” ; it may also be interpreted
in an infinite number of other ways. What all the ways have in
common is that they satisfy the axioms of geometry.

We often have, both in pure and applied mathematics, collec-
tions of formulae all logically deducible from a small number of
initial formulae, which may be called “axioms”. These axioms
may be regarded as hostages for the whole system, and we may
concentrate our attention exclusively upon them. The axioms
consist partly of terms having a known definition, partly of terms
which, in any interpretation, will remain variables, and partly
of terms which, though as yet undefined, are intended to acquire
definitions when the axioms are “interpreted**. The process of
interpretation consists in finding a constant signification for this
class of terms. The signification may be given by a verbal defini-
tion, or may be given ostensively. It must be such that, with this
interpretation, the axioms become true. (Before interpretation,
they are neither true nor false.) It thus follows that all their
consequences are also true.

Suppose, for example, we wish to interpret the formulae of
arithmetic. In Peano*s five axioms (given above) there are: first,
logical terms, such as “is a** and “is identical with**, of which
the meaning is supposed known ; second, variables, such as a and s,
which are to remain variables after interpretation; third, the terms
“o”, “number**, and “successor of**, for which an interpretation
is to find a constant meaning which makes the five axioms true.
As we saw, there are an infinite number of interpretations satis-
fying these conditions, but there is only one among them which
also satisfies empirical statements of enumeration, such as “I have
io fingers’*. In this case, therefore, there is one interpretation
which is very much more convenient than any of the others.

As we saw in the case of geometry, a given set of axioms may

257 R

HUMAN knowledge: its scope and limits

be capable of two sorts of interpretation, one logical and one
empirical. All nominal definitions, if pushed back far enough, must
lead ultimately to terms having only ostensive definitions, and
in the case of an empirical science the empirical terms must
depend upon terms of which the ostensive definition is given in
perception. The astronomer’s sun, for instance, is very different
from what we see, but it must have a definition derived from the
ostensive definition of the word “sun” which we learnt in child-
hood. Thus an empirical interpretation of a set of axioms, when
complete, must always involve the use of terms which have an
ostensive definition derived from sensible experience. It will not,
of course, contain only such terms, for there will always be logical
terms; but it is the presence of terms derived from experience
that makes an interpretation empirical.

The question of interpretation has been unduly neglected. So
long as we remain in the region of mathematical formulae, every-
thing appears precise, but when we seek to interpret them it
turns out that the precision is partly illusory. Until this matter
has been cleared up, we cannot tell with any exactitude what any
given science is asserting.

Chapter II

MINIMUM VOCABULARIES

I n the present chapter we shall be concerned with a linguistic
technique which is very useful in the analysis of scientific
concepts. There are as a rule a number of ways in which the
words used in a science can be defined in terms of a few among
them. These few may have ostensive definitions, or may have
nominal definitions in terms of words not belonging to the science
in question, or — so long as the science is not “interpreted” in
the sense considered in the last chapter — they may be left without
either ostensive or nominal definition, and regarded merely as
a set of terms having the properties which the science ascribes
to its fundamental terms. Such a set of initial words I call a
“minimum vocabulary” for the science in question, provided
that (a) every other word used in the science has a nominal
definition in terms of these words, and ( b ) no one of these initial
words has a nominal definition in terms of the other initial
words.

Everything said in a science can be said by means of the words
in a minimum vocabulary. For whenever a word occurs which
has a nominal definition, we can substitute the defining phrase;
if this contains words with a nominal definition, we can again
substitute the defining phrase, and so on, until none of the re-
maining words have nominal definitions. In fact, definable terms
are superfluous, and only undefined terms are indispensable. But
the question which terms are to be undefined is in part arbitrary.
Take, for example, the calculus of proposition, which is the
simplest and most completed example of a formal system. We
can take “or” and “not” as undefined, or “and” and “not”;
instead of two such undefined terms, we can take one, which
may be “not this or not that” or “not this and not that”. Thus
in general we cannot say that such-and-such a word must belong
to the minimum vocabulary of such-and-such a science, but at
most that there are one or more minimum vocabularies to which
it belongs.

Let us take geography as an example. I shall assume the voca-
bulary of geometry already established ; then our first distinctively

259

HUMAN knowledge: its scope and limits

geographical need is a method of assigning latitude and longitude.
For this it will suffice to have as part of our minimum vocabulary
“Greenwich”, “the North Pole”, and “West of”; but clearly any
other place would do as well as Greenwich, and the South Pole
would do as well as the North Pole. The relation “west of” is
not really necessary, for a parallel of latitude is a circle on the
earth’s surface in a plane perpendicular to the diameter passing
through the North Pole. The remainder of the words used in
physical geography, such as “land” and “water”, “mountain”
and “plain”, can now be defined in terms of chemistry, physics,
or geometry. Thus it would seem that it is the two words “Green-
wich” and “North Pole” that are needed in order to make
geography a science concerning the surface of the earth, and not
some other spheroid. It is owing to the presence of these two
words (or two others serving the same purpose) that geography
is able to relate the discoveries of travellers. It is to be observed
that these two words are involved wherever latitude and longitude
are mentioned.

As this example illustrates, a science is apt to acquire a smaller
minimum vocabulary as it becomes more systematic. The ancients
knew many geographical facts before they knew how to assign
latitudes and longitudes, but to express these facts they needed
a larger number of undefined words than we need. Since the
earth is a spheroid, not a sphere, “North-Pole” need not be
undefined : we can define the two Poles as the extremities of the
earth’s shortest diameter, and the North Pole as the Pole nearer
to Greenwich. In this way we can manage with “Greenwich” as
the only undefined term peculiar to geography. The earth itself
is defined as “that spheroid whose surface is formed of land and
water bounded by air, and on whose surface Greenwich is
situated”. But here we seem to reach a dead end in the way of
diminishing our minimum vocabulary: if we are to be sure that
we are talking about the earth, we must mention some place on
its surface or having a given geometrical relation to it, and the
place must be one which we can recognize. Therefore although
“New York” or “Moscow” or “Timbuctoo” would do just as
well as “Greenwich”, some place must be included in any minimum
vocabulary for geography.

One further point is illustrated by our discussion of Greenwich,
and that is, that the terms which are officially undefined in a

260

MINIMUM VOCABULARIES

science may not be identical with those that are undefined for
a given person. If you have never seen Greenwich, the word
“Greenwich” cannot, for you, have an ostensive definition; there-
fore you cannot understand the word unless it has a nominal
definition. In fact, if you live in a place called “P”, then for you
“P” takes the place of Greenwich, and your official longitude,
for you, defines the meridian of Greenwich, not the longitude
of P. Such considerations, however, are pre-scientific, and are
usually ignored in the analysis of scientific concepts. For certain
purposes, they cannot be ignored, particularly when we are con-
sidering the relation of science to sensible experience; but as a
rule there is little danger in ignoring them.

Let us consider next the question of minimum vocabularies
for astronomy. Astronomy consists of two parts, one a kind of
cosmic geography, the other an application of physics. Statements
as to the size and orbits of the planets belong to cosmic geography,
whereas Newton’s and Einstein’s theories of gravitation belong
to physics. The difference is that, in the geographical part, we
are concerned with statements of fact as to what is where, while
in the part which is physics we are concerned with laws. As I
shall presently be considering physics on its own account, let us
consider first the geographical part of astronomy. In this part,
so long as it is in an elementary stage, we need proper names
for the sun, the moon, the planets, and all the stars and
nebulae. The number of proper names required can, however,
be steadily reduced as the science of astronomy advances.
“Mercury” can be defined as meaning “the planet nearest the
sun”, “Venus” as “the second planet”, “the earth” as “the
third planet”, and so on. Constellations are defined by their
co-ordinates, and the several stars in a constellation by their
order of brightness.

On this system, “the sun” will remain part of our minimum
vocabulary, and we shall need what is necessary for defining
celestial co-ordinates. “The Pole Star” will not be necessary,
since it may be defined as “the star without diurnal revolution”,
but we shall need some other heavenly body to fulfil the function
which Greenwich fulfils in terrestrial geography. In this way
official astronomy could get on with (it would seem) only two
proper names, “the sun” and (say) “Sirius”. “The moon”, for
instance, can be defined as “the body whose co-ordinates on such-

261

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

and-such a date are so-and-so.” With this vocabulary we can,
in a sense, state everything that the astronomer wishes to say,
just as, with Peano’s three undefined terms, we can state all
arithmetic.

But just as Peano’s system proves inadequate when we come
to counting, so our official astronomy proves inadequate when
we attempt to link it to observation. There are two essential
propositions which it fails to include, namely “that is the sun”
and “that is Sirius”. We have, it appears, formed a vocabulary
for astronomy in the abstract, but not for astronomy as a record
of observations.

Plato, who was interested in astronomy solely as a body of
laws, wished it to be wholly divorced from sense; those who were
interested in the actual heavenly bodies that happen to exist
would, he said, be punished in the next incarnation by being
birds. This point of view is not nowadays adopted by men of
science, but it, or something very like it, is to be found in the
works of Carnap and some other logical positivists. They are
not, I think, conscious of holding any such opinion, and
would vehemently repudiate it; but absorption in words,
as opposed to what they mean, has exposed them to Platonic
temptation, and led them down strange paths towards perdition,
or what an empiricist must consider such. Astronomy is not
merely a collection of words and sentences; it is a collection of
words and sentences chosen, from others that were linguistically
just as good, because they described a world connected with
sensible experience. So long as sensible experience is ignored,
no reason appears for concerning ourselves with a large body
having just so many planets at just such distances from it. And
the sentences in which sensible experience breaks in are such
as “that is the sun”.

Every advanced empirical science has two aspects: on the one
hand, it consists of a body of propositions interconnected in
various ways, and often containing a small selection from which
all the others can be deduced ; on the other hand, it is an attempt
to describe some part or aspect of the universe. In the former
aspect, the truth or falsehood of the several propositions is not
in question, but only their mutual connections. For example, if
gravitation varied directly as the distance, planets (if any) would
revolve round the sun (if it existed) in ellipses of which the sun

262

MINIMUM VOCABULARIES

would occupy the centre, not a focus. This proposition is not
part of descriptive astronomy. There is a similar statement, also
not part of descriptive astronomy, saying that if gravitation varies
inversely as the square of the distance, planets (if any) will go
round the sun (if any) in ellipses of which the sun will occupy
a focus. This is different from the two statements: gravitation
varies inversely as the square of the distance, and planets revolve
in ellipses round the sun in a focus. The former statement is a
hypothetical; the two latter assert both the antecedent and the
consequent of the previous hypothetical. What enables them to
do this is the appeal to observation.

The appeal to observation is made in statements such as “that
is the sun”; such statements, therefore, are essential to the truth
of astronomy. Such statements never appear in any finished
exposition of an astronomical theory, but they do appear while
a theory is being established. For instance, after the eclipse
observations in 1919, we were told that the photographs of certain
stars appeared with such-and-such a displacement towards the
sun. This was a statement as to the positions of certain dots on
a photographic plate, as observed by certain astronomers at a
certain date; it was a statement not primarily belonging to
astronomy, but to biography, and yet it constituted the evidence
for an important astronomical theory.

The vocabulary of astronomy, it thus appears, is wider if we
consider it as a body of propositions deriving truth, or at least
probability, from observation, than it is if we treat it as a purely
hypothetical system whose truth or falsehood does not concern
us. In the former case we must be able to say “that is the sun”,
or something of the sort; in the latter case, no such necessity
arises.

Physics, which we must next consider, is in a different position
from geography and astronomy, since it is not concerned to say
what exists where, but only to establish general laws. “Copper
conducts electricity” is a law of physics, but “there is copper
in Cornwall” is a fact of geography. The physicist as such does
not care where there is copper, so long as there is enough in his
laboratory.

In the earlier stages of physics the word “copper” was neces-
sary, but now it has become definable. “Copper” is “the element
whose atomic number is 29”, and this definition enables us to

263

HUMAN KNOWLEDGE: 'ITS SCOPE AND LIMITS

deduce many things about the copper atom. All the elements can
be defined in terms of electrons and protons, or at any rate of
electrons, positrons, neutrons, and protons. (Perhaps a proton
consists of a neutron and a positron.) These units themselves
can be defined by their mass and electric charge. In the last
analysis, since mass is a form of energy, it would seem that energy,
electric charge, and space-time co-ordinates are all that physics
needs ; and owing to the absence of the geographical element the
co-ordinates can remain purely hypothetical, i.e. there need be
no analogue of Greenwich. Physics as a “pure” science — i.e. apart
from methods of verification — would seem, therefore, to require
only a four-dimensional continuum containing distributions of
varying amounts of energy and electricity. Any four-dimensional
continuum will do, and “energy” and “electricity” need only be
quantities whose mode of change of distribution is subject to
certain assigned laws.

When physics is brought to this degree of abstraction it becomes
a branch of pure mathematics, which can be pursued without
reference to the actual world, and which requires no vocabulary
beyond that of pure mathematics. The mathematics, however,
are such as no pure mathematician would have thought of for
himself. The equations, for instance, contain Planck’s constant A,
of which the magnitude is about 6*55 X io“ 27 erg secs. No one
would have thought of introducing just this quantity if there had
not been experimental reasons for doing so, and as soon as we
introduce experimental reasons the whole picture is changed. The
four-dimensional continuum is no longer a mere mathematical
hypothesis, but the space-time continuum to which we have been
led by successive refinements of the space and time with which
we are familiar in experience. Electricity is no longer just any
quantity, but the thing measured by the observable behaviour
of our electrical instruments. Energy, though highly abstract, is
a generalization arrived at by means of completely concrete
experiments such as those of Joule. Physics as verifiable, therefore,
uses various empirical concepts in addition to those purely abstract
concepts that are needed in “pure” physics.

Let us consider in more detail the definition of such a term
as “energy”. The important point about energy is its constancy,
and the chief step in establishing its constancy was the deter-
mination of the mechanical equivalent of heat. This was effected

264

MINIMUM VOCABULARIES

by observation, for example of thermometers. If, then, we mean
by “physics'* not merely the body of physical laws, but these
together with the evidence for their truth, then we must include
in “physics” Joule's perceptions when he looked at thermometers.
And what do we mean by “heat”? The plain man means a certain
kind of sensation, or its (to him) unknown cause; the physicist
means a rapid agitation of the minute parts of bodies. But what
has led the physicist to this definition? Only the fact that, when
we feel heat, there is reason to think that such agitation is occur-
ring. Or take the fact that friction causes heat: our primary
evidence for this fact is that when we have seen friction we can
feel heat. All the non-mathematical terms used in physics con-
sidered as an experimental science have their origin in our sensible
experience, and it is only on this account that sensible experience
can confirm or confute physical laws.

It thus appears that, if physics is regarded as a science based
on observation, not as a branch of pure mathematics, and if the
evidence for physical laws is held to be part of physics, then any
minimum vocabulary for physics must be such as to enable us
to mention the experiences upon which our physical beliefs are
based. We shall need such words as “hot”, “red”, “hard”, not
only to describe what physics asserts to be the condition of bodies
that give us these sensations, but also to describe the sensations
themselves. Suppose I say, for instance: By “red” light I mean
light of such-and-such a range of wave-lengths. In that case the
statement that light of such wave-lengths makes me see red is
a tautology, and until the nineteenth century people were uttering
meaningless noises when they said that blood is red, because
nothing was known of the correlation of wave-lengths with sen-
sations of colour. This is absurd. It is obvious that “red” has a
meaning independent of physics, and that this meaning is relevant
in collecting data for the physical theory of colours, just as the
pre-scientific meaning of “hot” is relevant in establishing the
physical theory of heat.

The main conclusion of the above discussion of minimum
vocabularies is that every empirical science, however abstract,
must contain in any minimum vocabulary words descriptive of
our experiences. Even the most mathematical terms, such as
“energy”, must, when the chain of definitions is completed until
we reach terms of which there is only an ostensive definition,

265

HUMAN knowledge: its scope and limits

be found to depend for their meaning upon terms directly
descriptive of experiences, or even, in what may be called the
“g e °g ra phi c al” sciences, giving names to particular experiences.
This conclusion, if valid, is important, and affords great assistance
in the work of interpreting scientific theories.

266

Chapter III
STRUCTURE

I n the present chapter we shall be concerned with a purely
logical discussion which is essential as a preliminary to any
further steps in the interpretation of science. The logical
concept which I shall endeavour to explain is that of “structure”.

To exhibit the structure of an object is to mention its parts
and the ways in which they are interrelated. If you were learning
anatomy, you might first learn the names and shapes of the various
bones, and then be taught where each bone belongs in the skeleton.
You would then know the structure of the skeleton in so far as
anatomy has anything to say about it. But you would not have
come to an end of what can be said about structure in relation
to the skeleton. Bones are composed of cells, and cells of mole-
cules, and each molecule has an atomic structure which it is the
business of chemistiy to study. Atoms, in turn, have a structure
which is studied in physics. At this point orthodox science ceases
its analysis, but there is no reason to suppose that further analysis
is impossible. We shall have occasion to suggest the analysis of
physical entities into structures of events, and even events, as
I shall try to show, may be regarded with advantage as having
a structure.

Let us consider next a somewhat different example of structure,
namely sentences. A sentence is a series of words, arranged in
order by the relation of earlier and later if the sentence is spoken,
and of left to right if it is written. But these relations are not
really between words; they are between instances of words. A
word is a class of similar noises, all having the same meaning
or nearly the same meaning. (For simplicity I shall confine myself
to speech as opposed to writing.) A sentence also is a class of
noises, since many people can utter the same sentence. We must
say, then, not that a sentence is a temporal series of words, but
that a sentence is a class of noises, each consisting of a series of
noises in quick temporal succession, each of these latter noises
being an instance of a word. (This is a necessary but not a sufficient
characteristic of a sentence ; it is not sufficient because some series
of words are not significant.) I will not linger on the distinction

267

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

between different parts of speech, but will go on to the next stage
in analysis, which belongs no longer to syntax, but to phonetics.
Each instance of a word is a complex sound, the parts being the
separate letters (assuming a phonetic alphabet). Behind the
phonetic analysis there is a further stage: the analysis of the
complex physiological process of uttering or hearing a single letter.
Behind the physiological analysis is the analysis of physics, and
from this point onward analysis proceeds as in the case of the
bones.

In the above account I passed hastily over two points that need
elucidation, namely that words have meaning and sentences have
significance. “Rain” is a word, but “raim” is not, though both are
classes of similar noises. “Rain is falling” is a sentence, but “rain
snow elephant” is not, though both are series of words. To define
“meaning” and “significance” is not easy, as we saw in discussing
the theory of language. The attempt is not necessary so long as
we confine ourselves strictly to questions of structure. A word
acquires meaning by an external relation, just as a man acquires
the property of being an uncle. No post mortem, however
thorough, will reveal whether the man was an uncle or not, and
no analysis of a set of noises (so long as everything external to
them is excluded) will show whether this set of noises has
meaning, or significance if the set is a series of what seem to
be words.

, The above example illustrates that an analysis of structure,
however complete, does not tell you all that you may wish to
know about an object. It tells you only what are the parts of the
object and how they are related to each other ; it tells you nothing
about the relations of the object to objects that are not parts or
components of it.

The analysis of structure usually proceeds by successive stages,
as in both the above examples. What are taken as unanalysed
units in one stage are themselves exhibited as complex structures
in the next stage. The skeleton is composed of bones, the bones
of cells, the cells of molecules, the molecules of atoms, the atoms
of electrons, positrons, and neutrons; further analysis is as yet
conjectural. Bones, molecules, atoms, and electrons may each be
treated, for certain purposes, as if they were unanalysable units
devoid of structure, but at no stage is there any positive reason
to suppose that this is in fact the case. The ultimate units

268

STRUCTURE

so far reached may at any moment turn out to be capable of
analysis. Whether there must be units incapable of analysis
because they are destitute of parts, is a question which there
seems no way of deciding. Nor is it important, since there is
nothing erroneous in an account of structure which starts from
units that are afterwards found to be themselves complex. For
example, points may be defined as classes of events, but that does
not falsify anything in traditional geometry, which treated points
as simples. Every account of structure is relative to certain units
which are, for the time being, treated as if they were devoid of
structure, but it must never be assumed that these units will not,
in another context, have a structure which it is important to
recognize.

There is a concept of “identity of structure” which is of great
importance in relation to a large number of questions. Before
giving a precise definition of this concept I will give some pre-
liminary illustrations of it.

Let us begin with linguistic illustrations. Suppose that, in any
given sentence, you substitute other words, but in a way which
still leaves the sentence significant; then the new sentence has
the same structure as the original one. Suppose, e.g., you start
with “Plato loved Socrates”; for “Plato” substitute “Brutus”, for
“loved” substitute “killed”, and for “ Socrates” substitute “Caesar”.
You thus arrive at the sentence “Brutus killed Caesar”, which has
the same structure as “Plato loved Socrates”. All sentences having
this structure are called “dyadic-relation sentences”. Similarly
from “Socrates is Greek” you could have passed to “Brutus is
Roman” without change of structure; sentences having this
structure are called “subject-predicate sentences”. In this way
sentences can be classified by their structure ; there are in theory
an infinite number of structures that sentences may have.

Logic is concerned with sentences that are true in virtue of
their structure, and that always remain true when other words are
substituted, so long as the substitution does not destroy signifi-
cance. Take, for example, the sentence: “If all men are mortal
and Socrates is a man, then Socrates is mortal.” Here we may
substitute other words for “Socrates”, “man”, and “mortal”,
without destroying the truth of the sentence. It is true that there
are other words in the sentence, namely “if-then” (which must
count as one word), “all”, “are”, “and’*, “is”, “a”. These words

269

HUMAN knowledge: its scope and limits

must not be changed. But these are “logical” words, and their
purpose is to indicate structure; when they are changed, the
structure is changed. (All this raises problems, but it is not
necessary for our present purpose to go into them.) A sentence
belongs to logic if we can be sure that it is true (or that it is false)
without having to know the meanings of any of the words except
those that indicate structure. That is the reason for the use of
variables. Instead of the above sentence about Socrates and man
and mortal, we say: “If all cl's are /?’s and x is an a, then x is a j8.”
Whatever x and a and ft may be, this sentence is true; it is true
in virtue of its structure. It is in order to make this clear that we
use and “a” and “0” instead of ordinary words.

Let us take next the relation of a district to a map of it. If the
district is small, so that the curvature of the earth can be neglected,
the principle is simple : east and west are represented by right and
left, north and south by up and down, and all distances are reduced
in the same proportion. It follows that from every statement about
the map you can infer one about the district, and vice versa. If
there are two towns, A and B, and the map is on the scale of an
inch to the mile, then from the fact that the mark “A” is ten
inches from the mark “B” you can infer that A is ten miles from
B, and conversely; and from the direction of the line from the
mark “A” to the mark “B” you can infer the direction of the line
from A to B. These inferences are possible owing to identity of
structure between the map and the district.

Now take a somewhat more complicated illustration: the
relation of a gramophone record to the music that it plays. It is
obvious that it could not produce this music unless there were a
certain identity of structure between it and the music, which can
be exhibited by translating sound-relations into space-relations,
or vice versa — e.g. what is nearer to the centre on the record
corresponds to what is later in time in the music. It is only because
of the identity of structure that the record is able to cause the
music. Very similar considerations apply to telephones, broad-
casting, etc.

We can generalize such examples so as to deal with the rela-
tions of our perceptual experiences to the external world. A
wireless set transforms electromagnetic waves into sound waves;
a human organism transforms sound waves into auditory sensa-
tions. The electromagnetic waves and the sound waves have a

270

STRUCTURE

certain similarity of structure, and so (we may assume) have the
sound waves and the auditory sensations. Whenever one complex
structure causes another, there must be much the same structure
in the cause and in the effect, as in the case of the gramophone
record and the music. This is plausible if we accept the maxim
“same cause, same effect” and its consequence, “different effects,
different causes”. If this principle is regarded as valid, we can
infer from a complex sensation or series of sensations the structure
of its physical cause, but nothing more, except that relations of
neighbourhood must be preserved, i.e. neighbouring causes have
neighbouring effects. This argument is one which needs much
amplification ; for the moment I am merely mentioning it by way
of anticipation, in order to show one of the important applications
of the concept of structure.

We can now proceed to the formal definition of “structure”.
It is to be observed that structure always involves relations: a
mere class, as such, has no structure. Out of the terms of a given
class many structures can be made, just as many different sorts of
houses can be made out of a given heap of bricks. Every relation
has what is called a “field”, which consists of all the terms that
halve the relation to something or to which something has the
re ation. Thus the field of “parent” is the class of parents and
children, and the field of “husband” is the class of husbands and
wives. Such relations have two terms, and are called “dyadic”.
There are also relations of three terms, such as jealousy and
“between”; these are called “triadic”. If I say “A bought B from
C for D pounds”, I am using a “tetradic” relation. If I say “A
minds B’s love for C more than D’s hatred of E”, I am using a
“pentadic” relation. To this series of kinds of relation there is no
theoretical limit.

Let us in the first instance confine ourselves to dyadic relations.
We shall say that a class a ordered by the relation R has the same
structure as a class £ ordered by the relation S, if to every term
in a some one term in fi corresponds, and vice versa, and if when
two terms in a have the relation R, then the corresponding terms
in /? have the relation S, and vice versa. We may illustrate by the
similarity between a spoken and a written sentence. Here the class
of spoken words in the sentence is a, the class of written words in
the sentence is j8, and if one spoken word is earlier than another,
then the written word corresponding to the one is to the left of

271

HUMAN knowledge: its scope and limits

the written word corresponding to the other (or to the right if the
language is Hebrew). It is in consequence of this identity of
structure that spoken and written sentences can be translated into
each other. The process of learning to read and write is the process
of learning which spoken word corresponds to a given written
word and vice versa.

A structure may be defined by several relations. Take, for
instance, a piece of music. One note may be earlier or later than
another, or simultaneous with it. One note may be louder than
another, or higher in pitch, or differing through a wealth or
poverty of harmonics. All the relations of this kind that are
musically relevant must have analogues in a gramophone record
if it is to give a good reproduction. In saying that the record must
have the same structure as the music, we are not concerned with
only one relation R between the notes of the music and one
corresponding relation S between the corresponding marks on
the record, but with a number of relations such as R and a number
of corresponding relations such as S. Some maps use different
colours for different altitudes; in that case, different positions on
the map correspond to different latitudes and longitudes, while
different colours correspond to different elevations. The identity
of structure in such maps is greater than in others; that is why
they are able to give more information.

The definition of identity of structure is exactly the same for
relations of higher orders as it is for dyadic relations. Given, for
example, two triadic relations R and S, and given two classes
a and /? of which a is contained in the field of R while j8 is con-
tained in the field of S, we shall say that a ordered by R has the
same structure as 0 ordered by S if there is a way of correlating
one member of a to one of /J, and vice versa, so that, if a v a 2 , a 3
are correlated respectively with b v b 2> fi 3 , if R relates a v a 2f a 3
(in that order), then S relates b v b 2 , b 3 (in that order), and vice
versa. Here, again, there may be several relations such as R, and
several such as S; in that case, there is identity of structure in
various respects.

When two complexes have the same structure,* every statement
about the one, in so far as it depends only on structure, has a
corresponding statement about the other, true if the first was true,
and false if the first was false. Hence arises the possibility of a
dictionary, by means of which statements about the one complex

272

STRUCTURE

can be translated into statements about the other. Or, instead of a
dictionary, we may continue to use the same words, but attach
different meanings to them according to the complex to which
they are referred. This sort of thing happens in interpreting a
sacred text or the laws of physics. The “days” in the Biblical
account of the Creation are taken to mean “ages”, and in this way
Genesis is reconciled with geology. In physics, assuming that our
knowledge of the physical world is only as to the structure resulting
from the empirically known relation of “neighbourhood” in the
topological sense, we have immense latitude in the interpretation
of our symbols. Every interpretation that preserves the equations
and the connection with our perceptive experiences has an equal
claim to be regarded as possibly the true one, and may be used with
equal right by the physicist to clothe the bare bones of his mathe-
matics.

Take, for example, the question of waves versus particles. Until
recently it was thought that this was a substantial question : light
must consist either of waves or of little packets called photons.
It was regarded as unquestionable that matter consisted of
particles. But at last it was found that the equations were the same
if both matter and light consisted of particles, or if both consisted
of waves. Not only were the equations the same, but all the
verifiable consequences were the same. Either hypothesis, there-
fore, is equally legitimate, and neither can be regarded as having
a superior claim to truth. The reason is that the physical world
can have the same structure, and the same relation to experience,
on the one hypothesis as on the other.

Considerations derived from the importance of structure show
that our knowledge, especially in physics, is much more abstract
and much more infected with logic than it used to seem. There is
however a very definite limit to the process of turning physics into
logic and mathematics; it is set by the fact that physics is an
empirical science, depending for its credibility upon relations to
our perceptive experiences. The further development of this
theme must be postponed until we come to the theory of scientific
inference.

2 73

Chapter IV

STRUCTURE AND MINIMUM VOCABULARIES

T he reader will remember that, in relation to a given body
of knowledge, a minimum vocabulary is defined as one
having the two properties (i) that every proposition in the
given body of knowledge can be expressed by means of words
belonging to the minimum vocabulary, (2) that no word in this
vocabulary can be defined in terms of other words in it. In the
present chapter I wish to show the connection of this definition
with structure.

The first thing to notice is that a minimum vocabulary cannot
contain names for complexes of which the structure is known.
Take (say) the name “France”. This denotes a certain geographical
region, and can be defined as “all places within such-and-such
boundaries”. But we cannot conversely define the boundaries in
terms of “France”. We want to be able to say “this place is on
the boundaries of France”, which requires a name for this place,
or for constituents which compose it. “This place” enters into
the definition of “France”, but “France” does not enter into the
definition of “this place”.

It follows that every discovery of structure enables us to
diminish the minimum vocabulary required for a given subject-
matter. Chemistry used to need names for all the elements, but
now the various elements can be defined in terms of atomic
structure, by the use of two words, “electron” and “proton”
(or perhaps three words, “electron”, “positron”, and “neutron”).
Any region in space-time can be defined in terms of its parts, but
its parts cannot be defined in terms of it. A man can be defined
by enumerating, in the right temporal order, all the events that
happen to him, but the events cannot be defined in terms of him.
If you wish to speak both about complexes and about the things
that are in fact their constituents, you can always achieve it with-
out names for the complexes, if you know their structure. In this
way analysis simplifies, systematizes, and diminishes your initial
apparatus.

The words required in an empirical science are of three sorts.
There are, first, proper names, which usually denote some con-

274

STRUCTURE AND MINIMUM VOCABULARIES

tinuous portion of space-time; such are “Socrates”, “Wales”,
“the sun”. Then there are words denoting qualities or relations;
instances of qualities are “red”, “hot”, “loud”, and instances of
relations are “above”, “before”, “between”. Then there are
logical words, such as “or”, “not”, “some”, “all”. For our present
purposes we may ignore logical words, and concentrate upon the
other two kinds.

It is usually taken for granted that the analysis of something
that has a proper name consists in dividing it into spatio-temporal
parts. Wales consists of counties, the counties consist of parishes,
each parish consists of the church, the school, etc. The church
in turn has parts, and so we can continue (it is thought) until we
reach points. The odd thing is that we never do reach points, and
that the familiar building thus seems to be composed of an
infinity of unattainable and purely conceptual constituents. I
believe this view of spatio-temporal analysis to be mistaken.

Qualities and relations are sometimes analysable, sometimes
not. I do not believe that “before”, as we know it in experience,
can be analysed ; at any rate I do not know any analysis of it that
I am willing to accept. But in some cases the analysis of a relation
is obvious. “Grandparent” means “parent of parent”, “brother”
means “son of parent”, and so on. All family relationships can be
expressed by means of the three words “spouse”, “male”, and
“parent”; this is a minimum vocabulary in this subject-matter.
Adjectives (i.e. words denoting qualities) are often complex in
their meaning. Milton calls the woodbine “well-attired”, which
is a word of which the meaning is very complex. So is such a word
as “famous”. Words such as “red”, which come nearer to sim-
plicity, do not achieve it; there are many shades of red.

Whenever the analysis of a quality or relation is known, the
word for that quality or relation is unnecessary in our “basic
English”.

When we have words for every thing, quality, and relation that
we cannot analyse, we can express all our knowledge without
the need of any other words. In practice this would be too lengthy,
but in theory nominal definitions are unnecessary.

If the world is composed of simples, i.e. of things, qualities,
and relations that are devoid of structure, then not only all our
knowledge, but all that of Omniscience, could be expressed by
means of words denoting these simples. We could distinguish in

275

HUMAN knowledge: its scope and limits

the world a stuff (to use William James’s word) and a structure.
The stuff would consist of all the simples denoted by names,
while the structure would depend on relations and qualities for
which our minimum vocabulary would have words.

This conception can be applied without assuming that there is
anything absolutely simple. We can define as “relatively simple”
whatever we do not know to be complex. Results obtained by
using the concept of “relative simplicity” will still be true if
complexity is afterwards found, provided we have abstained from
asserting absolute simplicity.

If we allow denotative as opposed to structural definitions, we
can, at least apparently, content ourselves with a much smaller
apparatus of names. All places in space-time can be indicated by
their co-ordinates, all colours by their wave-lengths, and so on.
We have already seen that the assignment of space- time co-ordi-
nates requires a few proper names, say “Greenwich”, “the Pole
Star”, and “Big Ben”. But this is a very small apparatus compared
to names for all the different places in the universe. Whether this
way of defining spatio-temporal places enables us to say all that
we know about them, is a difficult question, to which I shall return
shortly. Before discussing it, it will be well to examine more
closely the questions that arise concerning qualities.

Consider the definition of the word “red”. We may define it
(i) as any shade of colour between two specified extremes in the
spectrum, or (2) as any shade of colour caused by wave-lengths
lying between specified extremes, or (3) (in physics) as waves
having wave-lengths between these extremes. There are different
things to be said about these three definitions, but there is one
thing to be said about all of them.

What is to be said about all of them is that they have an artificial,
unreal, and partly illusory precision. The word “red”, like the
word “bald”, is one which has a meaning that is vague at the edges.
Most people would admit that, if a man is not bald, the loss of
one hair will not make him so ; it follows by mathematical induc-
tion that the loss of all his hairs will not make him so, which is
absurd. Similarly, if a shade of colour is red, a very tiny change
will not make it cease to be red, from which it follows that all
shades of colour are red. The same sort of thing happens when
we use wave-lengths in our definition, since lengths cannot
be accurately measured. Given a length which, by the most

276

STRUCTURE AND MINIMUM VOCABULARIES

careful measurements, appears to be a metre, it will still appear
to be a metre if it is very slightly increased or diminished; there-
fore every length appears to be a metre, which again is absurd.

It follows from these considerations that any definition of “red”
which professes to be precise is pretentious and fraudulent.

We shall have to define “red”, or any other vague quality, by
some such method as the following. When the colours of the
spectrum are spread out before us, there are some that everybody
would agree to be red, and others that everybody would agree to
be not red, but between these two regions of the spectrum there
is a doubtful region. As we travel along this region, we shall begin
by saying “I am nearly certain that that is red”, and end by saying
“I am nearly certain that that is not red”, while in the middle
there will be a region where we have no preponderant inclination
either towards yes or towards no . All empirical concepts have this
character — not only obviously vague concepts such as “loud” or
“hot”, but also those that we are most anxious to make precise,
such as “centimetre” and “second”.

It might be thought that we could make “red” precise by
confining the term to those shades that we are certain are red.
This, however, though it diminishes the area of uncertainty, does
not abolish it. There is no precise point in the spectrum where
you are sure that you become uncertain. There will still be three
regions, one where you are certain that you are certain the shade
is red, one where you are certain that you are uncertain, and an
intermediate region where you are uncertain as to whether you
are certain or uncertain. And these three regions, like the previous
ones, will have no sharp boundaries. You have merely adopted
one of the innumerable techniques which diminish the area of
vagueness without ever wholly abolishing it.

The above discussion has proceeded on the assumption of
continuity. If all change is discrete — and we do not know that it
is not — then complete accuracy is theoretically possible. But if
there is discontinuity it lies, for the present, far below the level
of sensible discrimination, so that discreteness, even if it should
exist, would be useless as a help in defining empirically given
qualities.

Let us now ignore the problem of vagueness, and revert to our
three definitions. But we will now adapt them so as to be defini-
tions of a given shade of colour. This introduces no new diffi-

277

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

culties, since, as we have seen, the definition of “red” as a band
of colours requires a definition of the precise shades that form
its boundaries.

Let us suppose that I am seeing a certain coloured patch, and
that I call the shade of the patch “C”. Physics tells me that this
shade of colour is caused by light of wave-length A. I may then
define “C” as: (i) the shade of any patch that is indistinguishable
in colour from the patch I am seeing now ; or (2) as the shade of
any visual sensation caused by electromagnetic waves of wave-
length A; or (3) as electromagnetic waves of wave-length A. When
we are concerned only with physics, without regard to the methods
by which its laws are verified, (3) is the most convenient definition.
We use it when we speak of ultra-violet light, and when we say
that the light from Mars is red, and when, during a sunset, we
say that the sun’s light is not really red, but only looks red because
of intervening mist. Physics, per se y has nothing to say about
sensations, and if it uses the word “colour” (which it need not do),
it will wish to define it in a way that is logically independent of
sensation.

But although physics as a self-contained logical system does
not need to mention sensations, it is only through sensations that
physics can be verified. It is an empirical law that light of a certain
wave-length causes a visual sensation of a certain kind, and it is
only when such laws are added to those of physics that the total
becomes a verifiable system. The definition (2) has the defect of
concealing the force of the empirical law which connects wave-
length with sensation. Names for colours were used for thousands
of years before the undulatory theory of light was invented, and it
was a genuine discovery that wave-lengths grow shorter as we
travel along the spectrum from red to violet. If we define a shade
of colour by its wave-length, we shall have to add that sensations
caused by light of the same wave-length all have a recognizable
similarity, and that there is a lesser degree of similarity when the
wave-lengths differ, but only by a little. Thus we cannot express
all that we know on the subject without speaking about shades of
colour as known directly in visual sensation, independently of any
physical theory as to light-waves.

It would seem, therefore, that if we wish for clarity in exhibiting
the empirical data which lead us to accept physics, we shall do well
to adopt our first definition of a shade of colour, since we shall

278

STRUCTURE AND MINIMUM VOCABULARIES

certainly need some way of speaking about what this definition
defines, without having to make the detour through physics that
is involved in mentioning wave-lengths.

It remains, however, an open question whether the raw material
in our definitions of colours should be a given shade of colour
(wherever it may occur), or a given patch of colour, which can
only occur once. Let us develop both hypotheses.

Suppose I wish to give an account of my own visual field
throughout a certain day. Since we are concerned only with
colour, depth may be ignored. I have therefore at each moment
a two-dimensional manifold of colours. I shall assume that my
visual field can be divided into areas of finite size, within each
of which the colour is sensibly uniform. (This assumption is not
essential, but saves verbiage.) My visual field, on this assumption,
will consist of a finite number of coloured patches of varying
shape. I may start by giving a name to each patch, or by giving
a name to each shade of colour. We have to consider whether
there are any reasons for preferring one of these courses to the
other.

If I start by giving a name to each patch, I proceed to the
definition of a shade of colour by means of a relation of colour-
similarity between patches. This similarity may be greater or less;
we suppose that there is an extreme degree of it which may be
called “exact likeness”. This relation is distinguished by being
transitive, which is not the case with minor degrees of resem-
blance. For the reasons already given, we can never be sure that,
in any given case, there is exact colour-likeness between two
patches, any more than we can be sure that a given length is
exactly a metre. However, we can invent techniques which
approximate more and more closely to what would be needed for
establishing exact likeness.

We define the shade of colour of a given patch as the class of
patches having exact colour-likeness to it. Every shade of colour
is defined in relation to a “this”; it is “the shade of colour of this
patch”. To each “this”, as we become aware of it, we give a name,
say “P”; then “the shade of P” is defined as “all patches having
exact colour-likeness to P”.

The question now arises: given two patches whicfc are in-
distinguishable in colour, what makes me think them two? The
answer is obvious: difference of spatio-temporal position. But

279

HUMAN knowledge: its scope and limits

though this answer is obvious, it does not dispose of the problem.
For the sake of simplicity, let us suppose that the two patches are
parts of one visual field, but are not in visual contact with each
other. Spatial position in the momentary visual field is a quality,
varying according to distance from the centre of the field of
vision, and also according as the region in question is above or
below, to the right or to the left, of the centre. The various
qualities that small portions of the visual field may have are
related by relations of up-and-down, right-and-left. When we
move our eyes, the qualities associated with a given physical
object change, but if the various physical objects have not moved,
there will be no topological change in the part of the visual field
which is common to both occasions. This enables common sense
to ignore the subjectivity of visual position.

Concerning these visual positional qualities we have exactly
the same alternatives as in the case of shades of colour. We may
give a name to each quality, considered as something which is the
same on different occasions, or we may give a name to each
instance of the quality, and connect it with other instances of the
same quality by the relation of exact likeness. Let us concentrate
on the quality that distinguishes the centre of the field of vision,
and let us call this quality “centrality”. Then on one view there
is a single quality of centrality, which occurs repeatedly, while
on the other view there are many particulars which have exact
positional likeness, and the quality of centrality is replaced by
the class of these particulars.

When we now repeat, in relation to the particulars which are
instances of centrality, the question as to how we distinguish one
of these particulars from another, the answer is again obvious:
we distinguish them by their position in time. (There cannot be
two simultaneous instances of centrality in one person’s experi-
ence.) We must therefore now proceed to analyse difference of
position in time.

In regard to time, as in regard to space, we have to distinguish
objective and subjective time. Objective space is that of the
physical world, whereas subjective space is that which appears
in our percepts when we view the world from one place. So
objective time is that of physics and history, while subjective time
is that which appears in our momentary view of the world. In my
present state of mind there are not only percepts, but memories

280

STRUCTURE AND MINIMUM VOCABULARIES

and expectations; what I remember I place in the past, what I
expect I place in the future. But from the impartial stand-point
of history my memories and expectations are just as much now
as my percepts. When I remember, something is happening to
me now which, if I remember correctly, has a certain relation to
what happened at an earlier time, but what happened then is not
itself in my mind now. My memories are placed in a time-order,
just as my visual perceptions are placed in a space-order, by
intrinsic qualities, which may be called “degrees of remoteness”.
But however high a degree of remoteness a memory may possess,
it is still, from the objective historical point of view, an event
which is happening now.

I said a moment ago that there could not be two simultaneous
instances of centrality in one person’s experience, but in a certain
sense this may be false. If, when my eyes are open, I remember
some previous visual experience, there will be one instance of
centrality in my percepts and another in my memory, and these
are both now in historical time. But they are not both now in the
time of my present subjective experience. Thus the correct state-
ment is: two instances of centrality cannot be simultaneous in
historical time if they arc perceptual parts of one man’s experience,
and they cannot in any case be simultaneous in the subjective
time of a single experience composed of percepts and memories
and expectations.

There is a certain difficulty in the conception of a time which,
in a sense, is wholly now y and a space which, in a sense, is wholly
here. Yet these conceptions seem unavoidable. The whole of my
psychological space is here from the standpoint of physics, and
the whole of my psychological time is now from the standpoint
of history. Like Leibniz’s monads, we mirror the universe, though
very partially and very inaccurately ; in my momentary mirroring
there is a mirror-space and a mirror-time, which have a corre-
spondence, though not an exact one, with the impersonal space
and time of physics and history. From the objective standpoint,
the space and time of my present experience are wholly confined
within a small region of physical space-time.

We must now return from this digression to the question
whether we are to assume one quality of centrality 'which can
exist at various times, or a number of instances of it, each of which
exists only once. It begins to be obvious that the latter hypothesis

281

HUMAN knowledge: its scope and limits

will entail great unnecessary complications, which the former
hypothesis avoids. We can bring the question to a head by asking
what can be meant by “this”. Let us suppose “this” to be some
momentary visual datum. There is a sense in which it may be true
to say “I have seen this before”, and there is another sense in
which this cannot be true. If I mean by “this” a certain shade of
colour, or even a certain shade of a certain shape, I may have
seen it before. But if I mean something dated, such as might be
called an “event”, then clearly I cannot have seen it before. Just
the same considerations apply if I am asked “do you see this any-
where else”? I may be seeing the same shade of colour somewhere
else, but if in the meaning of “this” I include position in visual
space, then I cannot be seeing it somewhere else. Thus what we
have to consider is spatio-temporal particularity.

If we take the view — which I think the better one — that a
given quality, such as a shade of colour, may exist in different
places and times, then what would otherwise be instances of the
quality become complexes in which it is combined with other
qualities. A shade of colour combined with a given positional
quality cannot exist in two parts of one visual field, because the
parts of the field are defined by their positional qualities. There is
a similar distinction in subjective time: the complex consisting
of a shade of colour together with one degree of remoteness
cannot be identical with the complex consisting of the same shade
of colour and another degree of remoteness. In this way “instances”
can be replaced by complexes, and by this replacement a great
simplification can be effected.

It results from the above discussion that a possible minimum
vocabulary for describing the world of my experience can be
constructed as follows. Names are given to all the qualities of
experiences, including such qualities of visual space and remem-
bered time as we have been considering. We also have to have
words for experienced relations, such as right-and-left in one
visual field, and earlier-and-later in one specious present. We do
not need names for space-time regions, such as “Socrates” or
“France”, because every space-time region can be defined as a
complex of qualities or a system of such complexes. “Events”,
which have dates and cannot recur, are capable of being regarded
as always complex; whatever we do not know how to analyse is
capable of occurring repeatedly in various parts of space-time.

282

STRUCTURE AND MINIMUM VOCABULARIES

When we pass outside our own experience, as we do in physics,
we need no new words. Definitions of things not experienced
must be denotational. Qualities and relations, if not experienced,
can only be known by means of descriptions in which all the
constants denote things that are experienced. It follows that a
minimum vocabulary for what we experience is a minimum
vocabulary for all our knowledge. That this must be the case is
obvious from a consideration of the process of ostensive definition.

283

Chapter V

TIME, PUBLIC AND PRIVATE

T he purpose of this Part is to provide possible interpre-
tations of the concepts of science, in terms of possible
minimum vocabularies. It will not be asserted that no other
interpretations are possible, but it is hoped that, in the course
of the discussion, certain common characteristics of all acceptable
interpretations will emerge. In the present chapter we shall be
concerned to interpret the word “time”.

Most people will be inclined to agree with St. Augustine:
“What, then, is time? If no one asks of me, I know: if I wish to
explain to him who asks, I know not.” Philosophers, of course,
have learned to be glib about time, but the rest of mankind,
although the subject feels familiar, are apt to be aware that a few
questions can reduce them to hopeless confusion. “Does the
past exist? No. Does the future exist? No. Then only the present
exists? Yes. But within the present there is no lapse of time?
Quite so. Then time does not exist ? Oh I wish you wouldn’t be
so tiresome.” Any philosopher can elicit this dialogue by a suitable
choice of interlocuter.

Sir Isaac Newton, who understood the Book of Daniel, also
knew all about time. Let us hear what he has to say on the subject
in the Scholium following the initial definitions in the Principia:

“I do not define time, space, place and motion, as being well
known to all. Only I must observe, that the vulgar conceive
those quantities under no other notions but from the relation
they bear to sensible objects. And thence arise certain prejudices,
for the removal of which, it will be convenient to distinguish
them into absolute and relative, true and apparent, mathematical
and common. Absolute, true, and mathematical time, of itself,
and from its own nature flows equably without regard to anything
external, and by another name is called duration : relative, appar-
ent, and common time, is some sensible external (whether accurate
or unequable) measure of duration by the means of motion,
which is commonly used instead of true time; such as an hour,
a day, a month, a year.”

He goes on to explain that days are not all of equal length, and

284

TIME, PUBLIC AND PRIVATE

that perhaps there is nowhere in nature a truly uniform motion,
but that we arrive at absolute time, in astronomy, by correction
of “vulgar” time.

Sir Isaac Newton's “absolute” time, although it remained
embedded in the technique of classical physics, was not generally
accepted. The theory of relativity has provided reasons, within
physics, for its rejection, though these reasons leave open the
possibility of absolute space-time. But before relativity Newton's
absolute time was already widely repudiated, though for reasons
which had nothing to do with physics. Whether, before rela-
tivity, these reasons had any validity is a question which I think
we shall find it worth while to examine.

Although Newton says that he is not going to define time
because it is well known, he makes it clear that only “vulgar”
time is well known, and that mathematical time is an inference.
In modern terms, we should rather call it an adjustment than an
inference. The process of arriving at “mathematical” time is
essentially as follows: there are a number of periodic motions —
the rotations and revolutions of the earth and the planets, the
tides, the vibrations of a tuning fork, the heart beats of a healthy
man at rest — which are such that, if one of them is assumed to be
uniform, all the others are approximately uniform. If we take one
of them, say the earth's rotation, as uniform by definition, we
can arrive at physical laws — notably the law of gravitation —
which explain the phenomena, and show why the other periodic
motions are approximately uniform. But unfortunately the laws
so established are only approximate, and, what is more, they show
that the earth’s rotation should suffer retardation by tidal friction.
This is self-contradictory if the earth's rotation is taken as the
measure of time; we therefore seek a different measure, which
shall also make our physical laws approximate more nearly to
exact truth. It is found convenient not to take any actual motion
as defining the measure of time, but to adopt a compromise
measure which makes physical laws as accurate as possible.
It is this compromise measure that serves the purposes for which
Newton invoked “absolute” time. There is no reason, however,
to suppose that it represents a physical reality, for the choice of
a measure of time is conventional, like the choice between the
Christian and the Mohammedan eras. We choose, in fact, the
measure which gives the greatest attainable simplicity to the

285

human knowledge: its scope and limits

statement of physical laws, but we do so on grounds of conveni-
ence, not because we think that this measure is more “true” than
any other.

A frequent ground of objection to Newton’s “absolute” time
has been that it could not be observed. This objection, on the
face of it, comes oddly from men who ask us to believe in electrons
and protons and neutrons, quantum transitions in atoms, and
what not, none of which can be observed. I do not think that
physics can dispense with inferences that go beyond observation.
The fact that absolute time cannot be observed is not, by itself,
fatal to the view that it should be accepted; what is fatal is the
fact that physics can be interpreted without assuming it. When-
ever a body of symbolic propositions which there is reason to
accept can be interpreted without inferring such-and-such
unobserved entities, the inference from the body of propositions
in question to these supposed entities is invalid, since, even if
there are no such entities, the body of propositions may be true.
It is on this ground, and not merely because “absolute” time
cannot be observed, that Newton was mistaken in inferring it
from the laws of physics.

While the rejection of Newton’s view is a commonplace, few
people seem to realize the problems that it raises. In physics
there is an independent variable /, the values of which are
supposed to form a continuous series, and each to be what is
commonly called an “instant”. Newton regarded an instant as a
physical reality, but the modern physicist does not. Since, how-
ever, he continues to use the variable t , he must find some
interpretation for its values, and the interpretation must serve
the technical purposes that were served by Newton’s “absolute”
time. This problem of the interpretation of is the one that
concerns us in this chapter. In order to simplify the approach to
it, we will at first ignore relativity and confine ourselves to time
as it appears in classical physics.

We shall continue to give the name “instant” to a value of
the variable t 9 but we shall look for an interpretation of the word
“instant” in terms of physical data, that is to say, we shall expect
the word to have a definition, and not to belong to a minimum
physical vocabulary. All that we require of the definition is that
instants, so defined, should have the formal properties demanded
of them by mathematical physics.

286

TIME, PUBLIC AND PRIVATE

In seeking a definition of “instant” or “point”, the material
to be used depends upon the theory we adopt as to “particulars”
or proper names. We may take the view that when, for instance,
a given shade of colour appears in two separated locations, there
are two separate “particulars”, each of which is an “instance”
of the shade of colour, and is a subject of which qualities can
be predicated, but which is not defined by its qualities, since
another precisely similar particular might exist elsewhere. Or
we may take the view that a “particular” is a bundle of coexisting
qualities. The discussions of the preceding chapter, as well as
the earlier discussion of proper names, inclined us to the latter
view. I shall, however, in this and the two following chapters,
hypothetically adopt the former view, and in Chapter VIII I
shall show how to interpret what has been said in terms of the
latter view. For the moment, therefore, I take as raw material
“events”, which are to be imagined as each occupying a finite
continuous portion of space-time. It is assumed that two events
can overlap, and that no event recurs.

It is clear that time is concerned with the relation of earlier
and later; it is generally held also that nothing of which we
have experience has a merely instantaneous existence. Whatever
is earlier or later than something else I shall call an “event”.
We shall want our definition of “instant” to be such that an
event can be said to exist “at” certain instants and not at certain
others. Since we have agreed that events, so far as known to us,
are not merely instantaneous, we shall wish to define “instant”
in such a way that every event exists at a continuous stretch of
the series of instants. That instants must form a series defined
by means of the relation of earlier and later is one of the requisites
that our definition must fulfil. Since we have rejected Newton's
theory, we must not regard instants as something independent
of events, which can be occupied by events as hats occupy hat-
pegs. We are thus compelled to search for a definition which
makes an instant a structure composed of a suitable selection of
events. Every event will be a member of many such structures,
which will be the instants during which it exists: it is “at” every
instant which is a structure of which the event is a member.

A date is fixed with complete precision if it is known concerning
every event in the world whether it wholly preceded that date, or
will wholly come after it, or was in existence at that date. To this

287

HUMAN KNOWLEDGE: ITS SCOPE AND HjMtlTS

statement some one might object that, if the world were to remain
without change for (say) five minutes, there would be no way
of fixing a date within these five minutes if the above view were
adopted, for every event wholly preceding one part of the five
minutes would wholly precede every other part, every event
wholly subsequent to any part of the five minutes would be
wholly subsequent to every other part, and every event existing
at any part of the five minutes would exist throughout the whole
of them. This, however, is not an objection to our statement,
but only to the supposition that time could go on in an unchanging
world. On the Newtonian theory this would be possible, but on
a relational theory of time it becomes self-contradictory. If
time is to be defined in terms of events, it must be impossible for
the universe to be unchanging for more than an instant. And
when I say “impossible” I mean logically impossible.

Although we cannot agree with Newton that “time” does not
need to be defined, it is obvious that temporal statements demand
some undefined term. I choose the relation of earlier-and-
later, or of wholly-preceding. Between two events a and b three
temporal relations are possible: a may be wholly before b , or
b may be wholly before a , or a and b may overlap. Suppose
you wish to fix as accurately as possible some date within the
duration of a. If you say that your date is also to be within the
duration of b , you fix the date somewhat more accurately than
by merely saying that it is within the duration, of a , unless it
so happens that a and b both began and ended together. Suppose
now there is a third event c which overlaps with both a and b —
that is to say, in ordinary language (to which we are not yet
entitled), there is a period of time during which a and b and c
all exist. This period, in general, will be shorter than that during
which both a and b exist. We now look for a fourth event d which
overlaps with a and b and c y i.e., in ordinary language, exists
during some part of the time during which a and b and c all
exist; the time during which a and b and c and d all exist is,
in general, shorter than that during which any three of them
all exist. In this way, step by step, we get nearer to an exact date.

Let us suppose this process carried on as long as possible, i.e.
until there is no event remaining which overlaps with all the
events already in our group. I say that, when this stage has been
reached, the group of events that has been constructed may

288

TIME, PUBLIC AND PRIVATE

be defined as an “instant”. To prove that this assertion is legiti-
mate I only have to show that “instants”, so defined, have the
mathematical properties that physics demands. I do not have
to show that this is what people commonly mean when they speak
of “instants”, though it might be desirable to complete the argu-
ment by showing that they commonly mean nothing.

An “instant”, as I propose to define the term, is a class of
events having the following two properties: (i) all the events
in the class overlap; (2) no event outside the class overlaps with
every member of the class. This group of events, as I shall show,
does not persist for a finite time.

To sav that an event persists for a finite time can only mean,
on a relational view of time, that changes occur while it exists,
i.e. that the events which exist when it begins are not all identical
with the events existing when it ends. This amounts to saying
that there are events which overlap with the given event but
not with each other. That is to say: “0 lasts for a finite time”
means “there are two events b and c such that each overlaps
with a but b wholly precedes c\

We may apply the same definition to a group of events. If
the members of the group do not all overlap, the group as a whole
has no duration, but if they all overlap, we shall say that the
group as a whole lasts for a finite time if there are at least two
events which overlap with every member of the group although
one of them wholly precedes the other. If this is the case, change
occurs while the group persists; if not, not. Now if a group
constitutes an “instant” as above defined, no event outside the
group overlaps all the members of the group, and no event inside
the group wholly precedes any other event inside the group.
Therefore the group as a whole does not last for a finite time.
And therefore it may suitably be defined as an “instant”.

Instants will form a series ordered by a relation defined in
terms of the relation “wholly preceding” among events. One
instant is earlier than another if there is a member of the first
instant which wholly precedes a member of the second, i.e. if
some event “at” the first instant wholly precedes some event
“at” the second instant. It will be observed that being “at” an
instant is the same thing as being a member of the class which is
the instant.

According to the above definition, it is logically impossible

289 T

HUMAN knowledge: its scope and limits

for the world to remain unchanging throughout a finite time. If
two instants differ, they are composed (at least in part) of different
members, and that means that some event existing at the one
instant does not exist at the other.

Our theory makes no assumption as to whether there are or
are not events that exist only at one instant. Such events, if any,
would have the characteristic that any two events overlapping
with them would overlap with each other. In general, the “dura-
tion” of an event means “the class of those instants of which the
event in question is a member”. It is generally assumed that an
event occupies a continuous stretch of the series of instants;
this assumption, formally, is embodied in the “axiom” that
nothing wholly precedes itself. But this axiom is not necessary.

Something has already been said concerning the quantitative
measurement of time, but it may be well to re-state the view to
which we are led by physics. The quantitative measurement of
time is conventional except to this extent, that a larger measure
must be applied to a whole than to a part. We must assign a
larger measure to a year than to any month in that year, but we
might, if convenient, assign to that year a smaller measure than
to a month in some other year. It turns out, however, that this
is not convenient. Historically, astronomers started with the
assumption that the day and the year were each of constant
length ; then it turned out that, if the sidereal day was constant,
the solar day was not, but the year was. If the sidereal day was
constant by definition, a large number of other periodic occur-
rences were approximately constant; this led to dynamical laws
which suggested that it would be more convenient to treat the
sidereal day as not exactly constant owing to tidal friction. The
laws could be formulated with any measure of time, but naturally
astronomers and physicists preferred the measure which made
the statement of the laws simplest. As this so very nearly agreed
with the “natural” measures of days and years, its conventional
character was not perceived, and it could be supposed that
what was being defined was Newton’s “true” or “mathematical”
time, believed to have physical reality.

I have been speaking so far as if there were, as used to be
thought, one cosmic time for the whole universe. Since Einstein,
we know that this is not the case. Each piece of matter has its
own local time. There is very little difference between the local

290

TIME, PUBLIC AND PRIVATE

time of one piece of matter and that of another unless their
relative velocity is an appreciable fraction of the velocity of light.
The local time of a given piece of matter is that which will be
shown by a perfectly accurate chronometer which travels with it.
Beta-particles travel with velocities that do not fall very far
short of that of light. If we could place a chronometer on a beta-
particle, and make the particle travel in a closed path, we should
find, when it returned, that the chronometer would not agree
with one that had remained throughout stationary in the
laboratory. A more curious illustration (which I owe to Professor
Reichenbach) is connected with the possibility of travel to the
stars. Suppose we invented a rocket apparatus which could
send a projectile to Sirius with a velocity ten elevenths of that of
light. From the point of view of the terrestrial observer the
journey would take about 55 years, and one might therefore
suppose that if the projectile carried passengers who were young
when they started, they would be old when they arrived. But
from their point of view the journey will only have taken about
11 years. This will not only be the time taken as measured by
their clocks, but also the time as measured by their physiological
processes — decay of teeth, loss of hair, etc. If they looked and
felt like men of 20 when they started, they will look and feel like
men of 31 when they arrive. It is only because we do not habitually
come across bodies travelling with a speed approaching that of
light that such odd facts remain unnoticed except by men of
science.

If two pieces of matter (say the earth and a comet) meet and
part and meet again, and if in the interval their relative velocity
has been very great, the physicists (if any) who live on the two
pieces of matter will form different estimates of the lapse of
time between the two meetings, but they will agree as to which
of the two meetings was the earlier and which the later. “Earlier”
and “later”, therefore, as applied to two events happening to one
piece of matter, have no ambiguity: if there are several pieces
of matter to which both the given events happen, one of the
events will be earlier for all of them, and the other will be later
for all of them.

The construction of “instants” as classes of events, given
above, is to be held, for the present, as applying only to events
happening to one piece of matter — primarily the body of a given

291

HUMAN knowledge: its scope and limits

observer. The extension to cosmic time, which can be made in
many ways, all equally legitimate, is a matter which I shall not
deal with at present.

Instead of basing our construction on the events happening
to a given body, we may base it on those happening to a given
mind or forming part of a given experience. If the mind is mine,
I can experience occurrences of the sort expressed by the words
“A wholly precedes B”, for example, when I am listening to
successive strokes of a clock striking the hour. If A is an event
that I experience, everything that overlaps with A or wholly
precedes A or wholly succeeds A will constitute “my” time, and
only events belonging to “my” time will be involved in the
construction of “instants” belonging to “my” time . 1 The linking
up of my time with yours will thus remain a problem to be con-
sidered.

We may define a “biography” as a collection of events such
that, of any two, either they overlap or there is one that wholly
precedes the other. For the present I shall assume that, when a
biography has a psychological definition, it also has a physical
definition — i.e. the time-series constituted out of events that I
experience is identical with the time-series constructed out of
events that happen to my brain, or some part of it. Accordingly,
I shall speak of the “biography” of a piece of matter, not only
of the “biography” connected with some person’s experience.

What has been said so far can now be summed up in a series
of definitions.

An “event” is something which precedes or follows or overlaps
something.

The “biography” to which an event belongs is all the events
that it precedes or follows or overlaps.

An “instant” is a collection of events belonging to one bio-
graphy, and having the two properties that (a) any two events in
the collection overlap, ( b ) no event outside the collection over-
laps with all the members of the collection.

An event is said to “exist at” an instant if it is a member of
that instant.

One instant is said to be “earlier” than another if there is an
event at the one which wholly precedes some event at the other.

i “My” time in the above sense is not to be confused with the sub-
jective time of Part III, Chapter V.

292

TIME, PUBLIC AND PRIVATE

A “time-series of a given instant” is a series of instants, of
which the given instant is one, and having the property that,
of any two, one is earlier than the other.

A “time-series” is a time-series of some instant.

It is not assumed that an instant can only belong to one time-
series, nor that an event can only belong to one biography. But
it is assumed that, if a wholly precedes 6, then a and b are not
identical. This is an assumption which we shall have to examine,
and perhaps modify, at a later stage.

As the above construction of time-series is the simplest example
of a kind of procedure which will be frequently employed, I
shall spend a few moments in setting forth the reasons for its
adoption.

We start from the fact that, although physicists reject Newton’s
absolute time, they continue to employ the independent variable
t, of which the values are said to be “instants”. The values of t
are held to form a series ordered by a relation called “earlier-
and-later”. It is held also that there are occurrences called
“events”, which include as a sub-class everything that we can
observe. There are two observable temporal relations among
events: they may overlap, as when I hear a clock striking while
I see its hands pointing to twelve o’clock; or one may precede
another, as when I still remember the previous stroke of the
clock while I am hearing the present stroke. These are the data
of our problem.

Now if we are to use the variable t without assuming Newton’s
absolute time, we must find a way of defining the class of values
of t 9 that is to say, “instants” must not form part of our minimum
vocabulary, which, so far as it is not merely that of logic, must
consist of words whose meaning is known by experience.

Definitions are of two sorts, which may be called respectively
“denotational” and “structural”. An example of a denotational
definition is “the tallest man in the United States”. This is
certainly a definition, since there must be one and only one
person to whom it applies, but it defines the man merely by his
relations. Generally, a “denotational” definition is one which
defines an entity as the only one having a certain relation, or
certain relations, to one or more known entities. On 'the other
hand, when what we want to define is a structure composed of
known elements, we can define it by mentioning the elements and

*93

HUMAN knowledge: its scope and limits

the relations constituting the structure; this is what I call a
“structural” definition. If what I am defining is a class, it may
be only necessary to mention the structure, since the elements
may be irrelevant. For example, I can define an “octagon” as
“a plane figure having eight sides”; this is a structural definition.
But I might also define it as “a polygon of which all known
examples are in the following places”, then giving a list. This
would be a “denotational” definition.

A denotational definition is not complete without a proof of
the existence of the object denoted. “The man over io feet
tall” is, in logical form, just as good as “the tallest man in the
United States”, but it probably denotes no one. “The square
root of 2” is a denotational definition, but until our own day
there was no proof that it denoted anything; now we know
that it is equivalent to the structural definition “the class of
those rationals whose squares are less than 2”, and thereby
the question of “existence” (in the logical sense) is solved. Owing
to the possible doubt about “existence”, denotational definitions
are often unsatisfactory.

In the particular case of our variable t , a denotational definition
is excluded by our rejection of absolute time. We must therefore
seek a structural definition. This implies that instants must have
a structure, and that the structure must be built out of known
elements. We have, as data of experience, the relations “over-
lapping” and “preceding”, and we find that by means of these
we can build structures having the formal properties that mathe-
matical physicists demand of “instants”. Such structures, there-
fore, fulfil all required purposes without the need of any ad hoc
assumption. This is the justification of our definitions.

294

Chapter VI

SPACE IN CLASSICAL PHYSICS

I N this chapter we shall be concerned with space as it appears
in classical physics. That is to say, we shall be concerned to
find an “interpretation” (not necessarily the only possible one)
for the geometrical terms used in physics. Much more complicated
and difficult problems arise in regard to space than in regard to
time. This is partly because of problems introduced by relativity.
For the present, however, we will ignore relativity, and treat
space as separable from time after the manner of pre-Einsteinian
physics.

For Newton, space, like time, was “absolute”, that is to say,
it consisted of a collection of points, each devoid of structure,
and each one of the ultimate constituents of the physical world.
Each point was everlasting and unchanging; change consisted in
its being “occupied” sometimes by one piece of matter, sometimes
by another, and sometimes by nothing. As against this view,
Leibniz contended that space was only a system of relations,
the terms of the relations being material and not merely geo-
metrical points. Although both physicists and philosophers
tended more and more to take Leibniz's view rather than
Newton’s, the technique of mathematical physics continued to
be Newtonian. In the mathematical apparatus, “space” is still
an assemblage of “points”, each defined by three co-ordinates,
and “matter” is an assemblage of “particles”, each of which
occupies different points at different times. If we are not to
agree with Newton in ascribing physical reality to points, this
system requires some interpretation in which “points” have a
structural definition.

I have used the word “physical reality”, which may be held
to savour too much of metaphysics. What I mean can be expressed,
in a form more acceptable to modern taste, by means of the
technique of minimum vocabularies. Given a collection of names,
it may happen that some of the things named have a structural
definition in terms of others; in that case, there will be & minimum
vocabulary not containing the names for which definitions can be
substituted. For example, every French human being has a

295

HUMAN knowledge: its scope and limits

proper name, and “the French nation” may also be regarded as
a proper name, but it is an unnecessary one, since we can say:
“the French nation” is defined as “the class consisting of the
following individuals (here follows the list)”. Such a method is
only applicable to finite classes, but there are other methods not
subject to this limitation. We can define “France” by its geo-
graphical boundaries, and then define “French” as “born in
France”.

To this process of substituting structural definitions for names
there are obviously limits in practice, and perhaps (though
this may be questioned) there are also limits in theory. Assuming,
for the sake of simplicity, that matter consists of electrons and
protons, we could, in theory, give a proper name to each electron
and each proton; we could then define an individual human
being by mentioning the electrons and protons composing his
body at various times; thus names for individual human beings
are theoretically superfluous. Speaking generally, whatever has a
discoverable structure does not need a name, since it can be
defined in terms of the names of its ingredients and the words
for their relations. On the other hand, whatever has no known
structure needs a name if we are to be able to express all our
knowledge concerning it.

It is to be observed that a denotational definition does not
make a name superfluous. E.g. “the father of Alexander the
Great” is a denotational definition, but does not enable us to
express the fact which contemporaries could have expressed
by “that is Alexander’s father”, where “that” functions as a
name.

When we deny Newton’s theory of absolute space, while
continuing to use what we call “points” in mathematical physics,
our procedure is only justified if there is a structural definition
of “point” and (in theory) of particular points. Such a definition
must proceed by methods similar to those that we employed in
defining “instants”. This, however, is subject to two provisos:
first, that our manifold of points is to be three-dimensional, and
second, that we have to define a point at an instant. To say that
a point P at one time is identical with a point Q at another time
is to say something which has no definite meaning except a
conventional one which depends upon a choice of material axes.
As this matter has to do with relativity, I shall not consider it

296

SPACE IN CLASSICAL PHYSICS

further at present, but shall confine myself to the definition of
points at a given instant, ignoring the difficulties connected with
the definition of simultaneity.

In what follows I lay no stress on the particular method of
constructing points that I have adopted. Other methods are
possible, and some of these may be preferred. What is important
is only that such methods can be devised. In defining instants,
we used the relation of “overlapping” in a temporal sense — a
relation which holds between two events when (in ordinary
language) there is a time during which both exist. In defining
points, we use the relation of “overlapping” in a spatial sense,
which is to subsist between two simultaneous events that (in
ordinary language) occupy the same region of space, in whole
or in part. It is to be observed that events, unlike pieces of matter,
are not to be thought of as mutually impenetrable. The impene-
trability of matter is a property which results tautologically from
its definition. “Events”, however, are only defined as terms not
assumed to possess a structure, and having spatial and temporal
relations such as belong to finite volumes and finite periods of
time. When I say “such as”, I mean “similar as regards logical
properties”. But “overlapping” is not itself to be defined logically;
it is an empirically known relation, having, in the construction
which I advocate, only an ostensive definition.

In a manifold of more than one dimension, we cannot construct
anything having the properties required of “points” by means
of a two-term relation of “overlapping”. As
the simplest illustration, let us take areas on ^ I B

a plane. Three areas A, B, C, on a plane may A t>Jr

each overlap with the other two, without there

being any region common to all three. In the
accompanying figure, the circle A overlaps with the rectangle B
and the triangle C, and B overlaps with C, but there is no region
common to A and B and C. The basis of our construction will
have to be a relation of three areas, not of two. We shall say that
three areas are “copunctual” when there is a region common to all
three. (This is an explanation, not a definition.)

We shall assume that the areas with which we are concerned
are all either circles, or such shapes as can result from circles
by stretching or compressing in a manner which leaves them oval.
In that case, given three areas A, B, C which are copunctual, and

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

a fourth area D such that A, B, D are copunctual and also A, C, D
and B, C, D, then A, B, C, D all have a common region.

We now define a group of any number of areas as “copunctual”
if every triad chosen out of the group is copunctual. A copunctual
group of areas is a “point” if it cannot be enlarged without
ceasing to be copunctual, i.e. if, given any area X outside the
group, there are in the group at least two areas A and B such that
A, B, X are not copunctual.

This definition is only applicable in two dimensions. In three
dimensions, we must start with a relation of copunctuality between
four volumes, and the volumes concerned must all be either
spheres or such oval volumes as can result from spheres by
continuous stretching in some directions and compressing in
others. Then, as before, a copunctual group of volumes is one
in which every four are copunctual, and a copunctual group is
a “point” if it cannot be enlarged without ceasing to be co-
punctual.

In n dimensions the definitions are the same, except that the
original relation of copunctuality has to be between n + 1 regions.

“Points” are defined as classes of events by the above
methods, with the tacit assumption that every event “occupies”
a more or less oval region.

“Events” are to be taken, in the present discussion, as the
undefined raw material from which geometrical definitions are
to be derived. In another context we may inquire as to what can
be meant by an “event”, and we may then be able to carry
analysis a step further, 1 but for the present we regard the manifold
of “events”, with their spatial and temporal relations, as empirical
data.

The way in which spatial order results from our assumptions
is somewhat complicated. I shall say nothing about it here, as
I have dealt with it in Analysis of Matter , where, also, there is
a much fuller discussion of the definition of “points” (Chapters
XXVIII and XXIX).

Something must be said about the metrical properties of space.
Astronomers, in their popular books, astound us first by telling
us how immensely distant many of the nebulae are, and then by
telling us that after all the universe is finite, being a three-
dimensional analogue of the surface of a sphere. But in their less
1 See Part II, Chapter III and Part IV, Chapter IV.

SPACE IN CLASSICAL PHYSICS

popular books they tell us that measurement is merely conven-
tional, and that we could, if we chose, adopt a convention which
would make the furthest known nebula in the northern hemisphere
nearer to us than the antipodes are. If so, the vastness of the
universe is not a fact, but a convenience. I think this is only
partially true, but to disentangle the element of convention in
measurement is by no means easy. Before attempting it, some-
thing must be said about measurement in its elementary forms.

Measurement, even of the distance to remote nebulae, is built
up from measurements of distances on the surface of the earth,
and terrestrial measurements start with the assumption that certain
bodies may be regarded as approximately rigid. If you measure
the size of your room, you assume that your foot-rule is not
growing appreciably longer or shorter during the process. The
ordnance survey of England determines most distances by tri-
angulation, but this process demands that there shall be at least
one distance which is measured directly. In fact, a base line on
Salisbury Plain was chosen, and was measured carefully in the
elementary way in which we measure the size of our room: a
chain, which we may take as by definition of unit length, was
repeatedly applied to the surface of the earth along a line as nearly
straight as possible. This one length having been determined
directly, the rest proceeds by the measurement of angles and by
calculation: the diameter of the earth, the distance of the sun
and moon, and even the distances of the nearer fixed stars, can
be determined without any further direct measurement of lengths.

But when this process is scrutinized it is found to be full of
difficulties. The assumption that a body is “rigid” has no clear
meaning until we have already established a metric enabling us
to compare lengths and angles at one time with lengths and angles
at another, for a “rigid” body is one which does not alter its shape
or size. Then again we need a definition of a “straight line”, for
all our results will be wrong if the base line on Salisbury Plain
and the lines used in triangulation are not straight. It seems,
therefore, that measurement presupposes geometry (to enable us
to define “straight lines”) and enough physics to give grounds
for regarding some bodies as approximately rigid, and for com-
paring distances at one time with distances at another. The
difficulties involved are formidable, but are concealed by assump-
tions taken over from common sense.

299

HUMAN knowledge: its scope and limits

Common sense assumes, roughly speaking, that a body is rigid
if it looks rigid. Eels do not look rigid, but steel bars do. On the
other hand, a pebble at the bottom of a rippling brook may look
as wriggly as an eel, but common sense nevertheless holds it
to be rigid, because common sense regards the sense of touch
as more reliable than the sense of sight, and if you wade across
the brook in bare feet the pebble feels rigid. Common sense, in
so thinking, is Newtonian: it is convinced that at each moment
a body intrinsically has a certain shape and size, which either are
or are not the same as its shape and size at another moment.
Given absolute space, this conviction has a meaning, but without
absolute space it is prima facie meaningless. There must, however,
be an interpretation of physics which will account for the very
considerable measure of success resulting from common-sense
assumptions.

As in the case of the measurement of time, three factors enter
in: first, an assumption liable to correction; second, physical laws
which, on this assumption, are found to be approximately true;
third, a modification of the assumption to make the physical laws
more nearly exact. If you assume that a certain steel rod, which
looks and feels rigid, preserves its length unchanged, you will find
that the distance from London to Edinburgh, the diameter of the
earth, and the distance of Sirius, are all nearly constant, but are
slightly less in warm weather than in cold. It will then occur to
you that it will be simpler to say that your steel rod expands with
heat, particularly when you find that this enables you to regard
the above distances as almost exactly constant, and, further, that
you can see the mercury in the thermometer taking up more space
in warm weather. You therefore assume that apparently rigid
bodies expand with heat, and you do so in order to simplify the
statement of physical laws.

Let us get clear as to what is conventional and what is physical
fact in this process. It is a physical fact that if two steel rods,
neither of which feels either hot or cold, look as if they were
of the same length, and if, then, you heat one by the fire and
put the other in snow, when you first compare them again the
one that has been by the fire looks slightly longer than the one
that has been in the snow, but when both have again reached the
temperature of your room this difference will have vanished. I
am here assuming pre-scientific methods of estimating tempera-

300

SPACE IN CLASSICAL PHYSICS

ture: a hot body is one that feels hot, and a cold body is one that
feels cold. As a result of such rough pre-scientific observations
we decide that the thermometer gives an exact measure of some-
thing which is measured approximately by our feelings of heat
and cold; we can then, as physicists, ignore these feelings and
concentrate on the thermometer. It is then a tautology that my
thermometer rises with an increase of temperature, but it is a
substantial fact that all other thermometers likewise do so. This
fact states a similarity between the behaviour of my thermometer
and that of other bodies.

But the element of convention is not quite as I have just stated
it. I do not assume that my thermometer is right by definition;
on the contrary, it is universally agreed that every actual ther-
mometer is more or less inaccurate. The ideal thermometer, to
which actual thermometers only approximate, is one which, if
taken as accurate, makes the general law of the expansion of bodies
with rising temperature as exactly true as possible. It is an
empirical fact that, by observing certain rules in making ther-
mometers, we can make them approximate more and more closely
to the ideal thermometer, and it is this fact which justifies the
conception of temperature as a quantity having, for a given body
at a given time, some exact value which is likely to be slightly
different from that shown by any actual thermometer.

The process is the same in all physical measurements. Rough
measurements lead to an approximate law ; changes in the measur-
ing instruments (subject to the rule that all instruments for
measuring the same quantity must give as nearly as possible the
same result) are found capable of making the law more nearly
exact. The best instrument is held to be the one that makes the
law most nearly exact, and it is assumed that an ideal instrument
would make the law quite exact.

This statement, though it may seem complicated, is still not
complicated enough. There is seldom only one law involved, and
very often the law itself is only approximate. Measurements of
different quantities are interdependent, as we have just seen in
the case of length and temperature, so that a change in the way
of measuring one quantity may alter the measure of another.
Laws, conventions, and observations are almost inextricably
intertwined in the actual procedure of science. The result of an
observation is usually stated in a form which assumes certain laws

301

HUMAN knowledge: its scope and limits

and certain conventions; if the result contradicts the network of
laws and conventions hitherto assumed, there may be considerable
liberty of choice as to which should be modified. The stock
instance is the Michelson-Morley experiment, where the simplest
interpretation was found to involve a radical change in temporal
and spatial measurements.

Let us now return to the measurement of distance. There are
a number of rough pre- scientific observations which suggest the
methods of measurement actually adopted. If you walk or bicycle
along a level road with what feels like constant exertion, you will
take roughly equal times for successive miles. If the road has
to be tarred, the amount of material required for one mile will
be about the same as that required for another. If you motor
along the road, the time taken for each mile will be about what
your speedometer would lead you to expect. If you base trigono-
metrical calculations on the assumption that successive miles are
equal, the results will be in close agreement with those obtained
directly by measurement. And so on. All this shows that the
numbers obtained by the usual processes of measurement have
considerable physical importance, and give a basis for many
physical and physiological laws. But these laws, once formulated,
give a basis for amending processes of measurement, and for
regarding the result of the amended processes as more “accurate”,
though in fact they are only more convenient.

There is, however, one element in the notion of “accuracy”
which is not merely convenient. We are accustomed to the axiom
that things that are equal to the same thing are equal to one
another. This axiom has a specious and deceptive appearance
of obviousness, in spite of the fact that the empirical evidence is
against it. You may find that, by the most delicate tests you can
apply, A is equal to B, and B to C, but A is noticeably unequal
to C. When this happens, we say that A is not really equal to B,
or B to C. Oddly enough, this tends to be confirmed when the
technique of measurement is improved. But the real basis of our
belief in the axiom is not empirical. We believe that equalitv
consists in possession of a common property. Two lengths arc
equal if they have the same magnitude , and it is this magnitude
that we seek to express when we measure. If we are right in this
belief, the axiom is logically necessary. If A and B have the same
magnitude, and B and C have the same magnitude, then, of

302

SPACE IN CLASSICAL PHYSICS

necessity, A and C have the same magnitude, provided that
nothing has more than one magnitude.

Although this belief in a magnitude, as a property which several
measurable things may have in common, obscurely influences
common sense in its conceptions as to what is obvious, it is not
a belief which we ought to accept until we have evidence of its
truth in the particular subject-matter concerned. The belief that
there is such a property of each of a set of terms is logically
equivalent to the belief that there is a transitive symmetrical
relation which holds between any two terms of the set. (This
equivalence is what I formerly called the “principle of abstrac-
tion’ \) Thus in maintaining that there is a set of magnitudes
called “distances”, what we are maintaining is this: Between any
one point-pair and any other, there is either a symmetrical tran-
sitive relation or an asymmetrical transitive relation. In the former
case, we say that the distance between the one pair of points is
equal to the distance between the other pair; in the latter case
we say that the first distance is less or greater than the second,
according to the sense of the relation. The distance between two
points may be defined as the class of point-pairs having to it the
relation of equidistance.

This is as far as we can carry the question of the measurement
of distance without going into the question of the definition of
straight lines, which we must now consider.

The straight line has its common-sense origin as an optical
concept. Some lines look straight. If a straight rod is held, end
on, against the eye, the part nearest the eye hides all the rest,
whereas if the rod is crooked some of it will appear round the
corner. There are of course also other common-sense reasons for
the concept of straight lines. If a body is rotated, there is a straight
line, the axis of rotation, which remains unmoved. If you are
standing up in the Underground, you can tell when it goes round
a curve by your tendency to overbalance one way or the other.
It is also possible, up to a point, to judge straightness by the
sense of touch ; blind men become almost as good at judging shapes
as men who can see.

In elementary geometry straight lines are defined as wholes;
their chief characteristic is that a straight line is determinate as
soon as two of its points are given. The possibility of regarding
distance as a straightforward relation between two points depends

303

HUMAN knowledge: its scope and limits

upon the assumption that there are straight lines. But in the
modern geometry developed to suit the needs of physics there
are no straight lines in the Euclidean sense, and “distance” is
only definite when the two points concerned are very close
together. When the two points are far apart, we must first decide
by what route we are to trave’ from the one to the other, and
then add up the many small distances along the route. The
“straightest” line between the two is the one that makes this
sum a minimum. Instead of straight lines we have to use “geo-
desics”, which are routes from one point to another that are shorter
than any slightly different routes. This destroys the simplicity
of the measurement of distances, which becomes dependent upon
physical laws. The resulting complications in the theory of
geometrical measurement cannot be dealt with without examining
more closely the connection of physical laws with the geometry
of physical space.

304

Chapter VII
SPACE-TIME

E verybody knows that Einstein substituted space-time for
space and time, but people unfamiliar with mathematical
physics have usually only a very vague conception as to
the nature of the change. As it is an important change in relation
to our attempts to conceive the structure of the world, I shall
try, in this chapter, to explain those parts of it that have philoso-
phical importance.

Perhaps the best starting-point is the discovery that “simul-
taneity” is ambiguous when applied to events in different places.
Experiments, especially the Michelson-Morley experiment, led
to the conclusion that the velocity of light is the same for all
observers, however they may be moving. This seemed, at first
sight, to be a logical impossibility. If you are in a train which
is moving at 30 miles an hour, and you are passed by a train
which is moving at 60 miles an hour, its speed relatively to you
will be 30 miles an hour. But if it is moving with the velocity
of light, its speed relatively to you will be the same as its speed
relatively to fixed points on the earth. Beta particles sometimes
move with speeds up to 90 per cent of the velocity of light, but
if a physicist could move with such a particle, and be passed by
a light-ray, he would still judge that the light was moving,
relatively to him, at the same rate as if he were at rest in relation
to the earth. This paradox is explained by the fact that different
observers, all equipped with perfect chronometers, will form
different estimates of time-intervals and different judgments as
to simultaneity in different places.

It is not difficult to see the necessity for such differences when
once it has been pointed out. Suppose an astronomer observes
an event in the sun, and notes the time of his observation; he
will infer that the event happened about eight minutes before his
observation, since that is the length of time that it takes light to
travel from the sun to the earth. But now suppose that the earth
were travelling very fast towards the sun or away from it. Unless
you already knew at what moment, by terrestrial time, the event
on the sun took place, you would not know how far the light had

305

HUMAN knowledge: its scope and limits

had to travel, and therefore your observation would not enable
you to know when the event in the sun had taken place. That
is tp say, there would be no definite answer to the question : what
eft^iits on earth were simultaneous with the solar event that you
had observed ?

From the ambiguity of simultaneity it follows that there is a
parallel ambiguity in the conception of distance. If two bodies
are in relative motion, their distance apart is continually changing,
and in pre-relativity physics there was supposed to be such a
quantity as their “distance at a given instant”. But if there is
ambiguity as to what is the same instant for the two bodies, there
is also ambiguity as to “distance at a given instant”. One observer
will form one estimate, and another will form another, and there
is no reason to prefer either estimate. In fact, neither time-
intervals nor spatial distances are facts independent of the move-
ments of the observer's body. There is a kind of subjectivity
about measurements of time and space separately — not a psycho-
logical but a physical subjectivity, since it affects instruments,
not only mental observers. It is like the subjectivity of the
camera, which takes a photograph from a certain point of
view. Photographs from other points of view would look
different, and no one among them would have a claim to special
accuracy.

There is, however, one relation between two events which is
the same for all observers. Formerly there were two, distance in
space and lapse of time, but now there is only one, which is called
“interval”. It is because there is only this one relation of interval,
instead of distance and lapse of time, that we have to substitute
the one concept of space-time for the two concepts of space and
time. But although we can no longer separate space and time,
there are still two kinds of interval, one space-like and the other
time-like. The interval is space-like if a light-signal, sent out by
the body on which one event occurs, reaches the body on which
the other event occurs after this other event has taken place.
(It is to be observed that there is no ambiguity about the time-
order of events on a given body.) It is time-like if a light-signal
sent out from one event reaches the body on which the other
event occurs before this other event has taken place. Since nothing
travels faster than light, we may say that the interval is time-like
when one event may have an effect upon the other, or upon

306

SPACE-TIME

something in the same space-time region as the other ; when this
is not possible, the interval is space-like.

In the special theory of relativity the definition of “interval*’
is simple; in the general theory it is more complicated.

In the special theory, suppose that an observer, treating himself
as motionless, judges the distance between two events to be r,
and the lapse of time between them to be t . Then if c is the
velocity of light, the square of the interval is

c 2 1 2 — r 2

if it is time-like, while if it is space-like it is

r 2 — c 2 t 2

It is usually simpler technically to take it as always one of these,
in which case the square of the other sort of interval is negative,
and the interval is imaginary.

When neither gravitation nor electromagnetic forces are in-
volved, it is found that the interval, as above defined, is the same
for all observers, and may therefore be regarded as a genuine
physical relation between the two events.

The general theory of relativity removes the above restriction
by introducing a modified definition of “interval”.

In the general theory of relativity there is no longer a definite
“interval” between distant events, but only between events that
are very near together. At a great distance from matter the formula
for interval approximates to that in the special theory, but else-
where the formula varies according to the nearness of matter. It
is found that the formula can be so adjusted as to account for
gravitation, assuming that matter which is moving freely moves
in a geodesic, i.e. chooses the shortest or longest route from any
one point to a neighbouring point.

It is assumed that, independently of interval, space- time points
have an order, so that, along any route, one point can be between
two others which are near it. For example, the interval between
two different points on one light-ray is zero, but the points still
have a temporal order: if a ray travels outward from the sun,
the parts near the sun are earlier than the parts farther from it.
The space-time order of events is presupposed in the assignment
of co-ordinates, for although this is to a great extent conventional
it must always be such that neighbouring points have co-ordinates
that do not differ much, and that, as points approach closer to

307

HUMAN knowledge: its scope and limits

each other, the difference between their co-ordinates approaches
zero as a limit.

If the physical world is held to consist of a four-dimensional
manifold of events, instead of a manifold of persistent moving
particles, it becomes necessary to find a way of defining what
is meant when we say that two events are part of the history of
one and the same piece of matter. Until we have such a definition,
“motion” has no definite meaning, since it consists in one thing
being in different places at different times. We must define a
“particle”, or material point, as a series of space-time points
having to each other a causal relation which they do not have
to other space- time points. There is no difficulty of principle about
this procedure. Dynamical laws are habitually stated on the
assumption that there are persistent particles, and are used to
decide whether two events A and B belong to the biography of
one particle or not. We merely retain the laws, and turn the
statement that A and B belong to the same biography into a
definition of a “biography”, whereas before it seemed to be a
substantial assertion.

This point perhaps needs some further explanation. Starting
from the assumption that there are persistent pieces of matter,
we arrive at physical laws connecting what happens to a piece
of matter at one time with what happens to it at another. (The
most obvious of such laws is the law of inertia.) We now state
these laws in a different way: we say that, given an event of a
certain kind in a certain small region of space-time, there will
be neighbouring events in neighbouring regions which will be
related to the given event in certain specific ways. We say that
a series of events related to each other in these specific ways is
to be called one piece of matter at different times. Thus matter
and motion cease to be part of the fundamental apparatus of
physics. What is fundamental is the four-dimensional manifold
of events, with various kinds of causal relations. There will be
relations making us regard the events concerned as belonging to
one piece of matter, others making us regard them as belonging
to different but interacting pieces of matter, others relating a
piece of matter to its “empty” environment (e.g. emission of
light), and yet others relating events that are both in empty space,
e.g. parts of one light- ray.

The collecting of events into series such as will secure the

308

SPACE-TIME

persistence of matter is only partially and approximately possible.
When an atom is pictured as a nucleus with planetary electrons,
we cannot say, after a quantum transition, that such-and-such'
an electron in the new state is to be identified with such-and-such
an electron in the old state. We do not even know for certain that
the number of electrons in the universe is constant. Mass is only
a form of energy, and there is no reason why matter should not
be dissolved into other forms of energy. It is energy, not matter,
that is fundamental in physics. We do not define energy ; we merely
discover laws as to the changes in its distribution. And these laws
are no longer such as to determine a unique result where atomic
phenomena are concerned, though macroscopic occurrences
remain statistically determinate with an enormously high degree
of probability.

The continuity of space-time, which is technically assumed in
physics, has nothing in its favour except technical convenience.
It may be that the number of space-time points is finite, and that
space- time has a granular structure, like a heap of sand. Provided
the structure is fine enough, there will be no observable phenomenon
to show that there is not continuity. Theoretically, there might be
evidence against continuity, but there could never be conclusive
evidence in its favour.

The theory of relativity does not affect the space and time of
perception. My space and time, as known in perception, are cor-
related with those that, in physics, are appropriate to axes that
move with my body. Relatively to axes tied to a given piece of
matter, the old separation of space and time still holds ; it is only
when we compare two sets of axes in rapid relative motion that
the problems arise which the theory of relativity solves. Since no
two human beings have a relative velocity approaching that of
light, comparison of their experiences will reveal no such dis-
crepancies as would result if aeroplanes could move as fast as
beta particles. In the psychological study of space and time,
therefore, the theory of relativity may be ignored.

3°9

Chapter VIII

THE PRINCIPLE OF INDIVIDUATION

1 shall discuss in this Chapter the modern form of a very
old problem, much discussed by the scholastics, but still, in
our day, far from being definitively solved. The problem, in
its broadest and simplest terms, is this: “How shall we define
the diversity which makes us count objects as two in a census? 0
We may put the same problem in words that look different,
e.g. “what is meant by a ‘particular’?” or “what sort of objects
can have proper names?”

Three views have been influentially advocated.

First: a particular is constituted by qualities; when all its
qualities have been enumerated, it is fully defined. This is the
view of Leibniz.

Second : a particular is defined by its spatio-temporal position.
This is the view of Thomas Aquinas as regards material substances.

Third: numerical diversity is ultimate and indefinable. This,
I think, would be the view of most modern empiricists if they
took the trouble to have a definite view.

The second of the above three theories is reducible to either
the first or the third according to the way in which it is interpreted.
If we take the Newtonian view, according to which there actually
are points, then two different points are exactly alike in all their
qualities, and their diversity must be that bare numerical diversity
contemplated in the third theory. If, on the other hand, we take
— as every one now does — a relational view of space, the second
theory will have to say: “If A and B differ in spatio-temporal
position, then A and B are two”. But here there are difficulties.
Suppose A is a shade of colour: it may occur in a number of
places and yet be only one. Therefore our A and B must not be
qualities, or, if they are, they must be qualities that never recur.
If they are not qualities or bundles of qualities, they must be
particulars of the sort contemplated in our third theory; if they
are qualities or bundles of qualities, it is the first of our three
theories that we are adopting. Our second theory, therefore, may
be ignored.

The construction of points and instants in our three preceding

310

THE PRINCIPLE OF INDIVIDUATION

Chapters used “events” as its raw material. Various reasons, of
which the theory of relativity has been the most influential, have
made this procedure preferable to one which, like Newton’s,
allows “points”, “instants”, and “particles” as raw material. It
has been assumed, in our constructions, that a single event may
occupy a finite amount of space-time, that two events may overlap
both in space and in time, and that no event can recur. That is
to say, if A wholly precedes B, A and B are not identical. We
assumed also that, if A wholly precedes B, and B wholly pre-
cedes C, then A wholly precedes C. “Events” were provision-
ally taken as “particulars” in the sense of our third theory. It was
shown that, if a raw material of this sort is admitted, space-time
points and space-time order can be constructed.

But we are now concerned with the problem of constructing
space-time points and space-time order when our first theory is
adopted. Our raw material will now contain nothing that cannot
recur, for a quality can occur in any number of separate places.
We have therefore to construct something that does not recur, and
until we have done so we cannot explain space-time order.

We have to ask ourselves: what is meant by an “instance”?
Take some definite shade of colour, which we will call “C”. Let
us assume that it is a shade of one of the colours of the rainbow,
so that it occurs wherever there is a rainbow or a solar spectrum.
On each occasion of its occurrence, we say that there is an
“instance” of C. Is each instance an unanalysable particular, of
which C is a quality ? Or is each instance a complex of qualities
of which C is one? The former is the third of the above theories;
the latter is the first.

There are difficulties in either view. Taking first the view that
an instance of C is an unanalysable particular, we find that we
encounter all the familiar difficulties connected with the traditional
notion of “substance”. The particular cannot be defined or recog-
nized or known ; it is something serving the merely grammatical
purpose of providing the subject in a subject-predicate sentence
such as “this is red”. And to allow grammar to dictate our
metaphysic is now generally recognized to be dangerous.

It is difficult to see how something so unknowable as such a
particular would have to be can be required for the interpretation
of empirical knowledge. The notion of a substance as a peg on
which to hang predicates is repugnant, but the theory that we

3 11

HUMAN KNOWLEDGE! ITS SCOPE AND LIMITS

have been considering cannot avoid its objectionable features. I
conclude, therefore, that we must, if possible, find some other
way of defining space-time order.

But when we abandon particulars in the sense which we have
just decided to reject, we are faced, as observed above, with the
difficulty of finding something that will not be repeated. A simple
quality, such as the shade of colour C, cannot be expected to
occur only once. We shall seek to escape this difficulty by con-
sidering a “complex” of qualities. What I mean will be most
easily understood if stated in psychological terms. If I see some-
thing and at the same time hear something else, my visual and
auditory experiences have a relation which I call “compresence”.
If at the same moment I am remembering something that hap-
pened yesterday and anticipating with dread a forthcoming visit
to the dentist, my remembering and anticipating are also “corn-
present” with my seeing and hearing. We can go on to form
the whole group of my present experiences and of everything
compresent with all of them. That is to say, given any group
of experiences which are all compresent, if I can find anything
else which is compresent with all of them I add it to the group,
and I go on until there is nothing further which is compresent
with each and all of the members of the group. I thus arrive at
a group having the two properties: [a) that all the members of
the group are compresent, (6) that nothing outside the group is
compresent with every member of the group. Such a group I
shall call a “complete complex of compresence”.

Such a complex I suppose to consist of constituents most of
which, in the natural course of events, may be expected to be
members of many other complexes. The shade of colour C, we
supposed, recurs every time anybody sees a rainbow distinctly.
My recollection may be qualitatively indistinguishable from a
recollection that I had yesterday. My apprehension of dental pain
may be just what I felt before my last visit to the dentist. All these
items of the complex of compresence may occur frequently, and
are not essentially dated. That is to say, if A is one of them, and
A precedes (or follows) B, we have no reason to suppose that
A and B are not identical.

Have we any reason, either logical or empirical, to believe that
a complete complex of compresence, aB a whole, cannot be
repeated? Let us, in the first place, confine ourselves to the

312

THE PRINCIPLE OF INDIVIDUATION

person’s experience. My visual field is very complex, though
probably not infinitely complex. Every time I move my eyes, the
visual qualities connected with a given object which remains
visible undergo changes : what I see out of the corner of my eyes
looks different from what is in the centre of my field of vision.
If it is true, as some maintain, that my memory is coloured by
my whole past experience, then it follows logically that my
total recollections cannot be exactly similar on two different
occasions; even if we reject this doctrine, such exact similarity
seems very improbable.

From such considerations I think we ought to conclude that
the exact repetition of my total momentary experience, which
is what, in this connection, I call a “complete complex of corn-
presence”, is not logically impossible, but is empirically so
exceedingly improbable that we may assume its non-occurrence.
In that case, a complete complex of compresence will, so far as
one person’s experience is concerned, have the formal properties
required of “events”, i.e.: if A, B, C are complete complexes of
compresence, then if A wholly precedes B, A and B are not
identical; and if B also wholly precedes C, then A wholly precedes
C. We thus have the requisites for defining the time-order in one
person’s experience.

This, however, is only part, and not the most difficult part, of
what we have to accomplish. We have to extend space-time order
beyond one person’s experience to the experiences of different
people and to the physical world. In regard to the physical world,
especially, this is difficult.

So long as we confine ourselves to one person’s experience, we
need only concern ourselves with time. But now we have also
to take account of space. That is to say, we have to find a definition
of “events” which shall insure that each event has, not merely
a unique temporal position, but a unique spatio-temporal position.

So long as we confine ourselves to experiences, there is no
fresh difficulty of a serious kind. It may be taken as virtually
certain, on empirical grounds, that my visual field, whenever my
eyes are open, is not exactly similar to that of any one else. If
A and B are looking simultaneously at the same scene, there are
differences of perspective ; if they change places, A will not see
exactly what B was seeing, because of differences of eyesight,
changes of lighting meanwhile, and so on. In short, the reasons

313

HUMAN knowledge: its scope and limits

for supposing that no total momentary experience of A is ever
exactly like some total momentary experience of B are of the
same kind as the reasons for supposing that no two total momen-
tary experiences of A are ever exactly alike.

This being granted, we can establish a spatial order among
percipients by means of the laws of perspective, provided there
is any physical object that all the percipients concerned are per-
ceiving. If there is not, a process by means of intermediate links
can reach the same result. There are of course complications and
difficulties, but they are not such as concern our subject at all
closely, and we may safely ignore them.

What can be said about the purely physical world is hypo-
thetical, since physics gives no information except as to structure.
But there are reasons for supposing that, at every place in physical
space-time, there is at every moment a multiplicity of occur-
rences, just as there is in a mind. “Compresence”, which I take
to have a merely ostensive definition, appears in psychology as
“simultaneity in one experience ,, ) but in physics as “overlapping
in space-time”. If, as I maintain, my thoughts are in my head,
it is obvious that these are different aspects of one relation. How-
ever, this identification is inessential to my present argument.

When I look at the stars on a clear night, each star that I see
has an effect on me, and has an effect on the eye before it has
an effect on the mind. It follows that, at the surface of the eye,
something causally connected with each visible star is happening.
The same considerations apply to ordinary objects seen in day-
light. At this moment I can see white pages covered with writing,
some books, an oval table, innumerable chimneys, green trees,
clouds, and blue sky. I can see these things because there is a
chain of physical causation from them to my eyes and thence
to the brain. It follows that what is going on at the surface of
my eye is as complex as my visual field, in fact as complex as
the whole of what I can see. This complexity must be physical,
not merely physiological or psychological; the optic nerve could
not make the complex responses that it does make except under
the influence of equally complex stimuli. We must hold that,
wherever the light of a certain star penetrates, something con-
nected with that star is happening. Therefore in a place where
a telescope photographs many millions of stars, many millions
of things must be happening, each connected with its own star.

THE PRINCIPLE OF INDIVIDUATION

These things are only "experienced” in places where there is a
recording nervous system, but that they happen in other places
also can be shown by cameras and dictaphones. There is therefore
no difficulty of principle in constructing "complexes of corn-
presence”, where there are no percipients, on the same principles
as we employed in dealing with momentary experiences.

Abandoning speculations about the physical world, about which
our knowledge is very limited, let us return to the world of
experience. The view which I am suggesting, as preferable to the
assumption of such wholly colourless particulars as points of space
or particles of matter, may be expressed as follows :

There is a relation, which I call "compresence”, which holds
between two or more qualities when one person experiences them
simultaneously — for example, between high C and vermilion when
you hear one and see the other. We can form groups of qualities
having the following two properties: (a) all members of the
group are compresent; ( b ) given anything not a member of the
group, there is at least one member of the group with which it
is not compresent. Any one such complete group of compresent
qualities constitutes a single complex whole, defined when its
constituents are given, but itself a unit, not a class. That is to
say, it is something which exists, not merely because its con-
stituents exist, but because, in virtue of being compresent, they
constitute a single structure. One such structure, when composed
of mental constituents, may be called a "total momentary
experience”.

Total momentary experiences, as opposed to qualities, have
time relations possessing the desired characteristics. I can see
blue yesterday, red to-day, and blue again to-morrow. Therefore,
so far as qualities are concerned, blue is before red and red is
before blue, while blue, since it occurs yesterday and to-morrow,
is before itself. We cannot therefore construct, out of qualities
alone, such a relation as will generate a series. But out of total
momentary experiences we can do this, provided no total momen-
tary experience ever exactly recurs. That this does not happen
is an empirical proposition, but, so far as our experience goes,
a well-grounded one. I regard it as a merit in the above theory
that it gets rid of what would otherwise be synthetic a priori
knowledge. That, if A precedes B, B does not precede A, and
that, if A precedes B and B precedes C, then A precedes C, are

315

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

synthetic propositions ; moreover, as we have just seen, they are
not true if A and B and C are qualities. By making such statements
(in so far as they are true) empirical generalizations, we overcome
what would otherwise be a grave difficulty in the theory of
knowledge.

I come back now to the conception of “instance”. An “instance”
of a quality, as I wish to use the word, is a complex of compresent
qualities of which the quality in question is one. In some cases
this view seems natural. An instance of “man” has other qualities
besides humanity: he is white or black, French or English, wise
or foolish, and so on. His passport enumerates enough of his
characteristics to distinguish him from the rest of the human
race. Each of these characteristics, presumably, exists in many
other instances. There are baby giraffes who have the height men-
tioned in his passport, and parrots who have the same birthday
as he has. It is only the assemblage of qualities that makes the
instance unique. Every man, in fact, is defined by such an
assemblage of qualities, of which humanity is only one.

But when we come to points of space, instants of time, particles
of matter, and such stock-in-trade of abstract science, we feel as
if a particular could be a “mere” instance, differentiated from
other instances by relations, not by qualities. To some degree,
we think this of less abstract objects: we say “as like as two
peas”, suggesting that between two peas there are no qualitative
differences. We think also that two patches of colour may be
merely two, and may differ only numerically. This way of thinking,
I maintain, is a mistake. I should say that, when the same shade
of colour exists in two places at once, it is one, not two; there
are, however, two complexes, in which the shade of colour is
combined with the qualities that give position in the visual field.
People have become so obsessed with the relativity of spatial
position in physics that they have become oblivious of the
absoluteness of spatial position in the visual field. At every
moment, what is in the centre of my field of vision has a quality
that may be called “centrality”; what is to the right is “dexter”,
what to the left “sinister”, what above “superior”, what below
“inferior”. These are qualities of the visual datum, not relations.
It is the complex consisting of one such quality combined with
a shade of colour that is distinct from the complex consisting of
the same shade elsewhere. In short, the multiplicity of instances

3 j 6

THE PRINCIPLE OF INDIVIDUATION

of a given shade of colour is formed exactly as the multiplicity
of instances of humanity is formed, namely by the addition of
other qualities.

As for points, instants, and particles, in so far as they are not
logical fictions similar considerations apply. Take first instants.
It will be found that what I call a “total momentary experience”
has all the formal properties required of an “instant” in my
biography. And it will be found that, where there is only matter,
the “complete complex of compresence” may serve to define an
instant of Einsteinian local time, or to define a “point-instant”
in cosmic space-time. Points in perceptual space are defined
without any trouble, since the qualities of up-and-down, right-
and-left, in their various degrees, have already all the properties
that we require of “points”. It is indeed this fact, together with
perception of depth, that has led us to place such emphasis on
the spatial characteristics of the world.

I do not think “particles” can be dealt with quite in the above
manner. In any case, they are no longer part of the fundamental
apparatus of physics. They are, I should say, strings of events
interconnected by the law of inertia. They are no longer indes-
tructible, and have become merely convenient approximations.

I come now to a possible objection to the above theory, which
was advanced by Arnauld against Leibniz. If a “particular” is
really a complex of qualities, then the statement that such-and-
such a particular has such-and-such a quality must, when true,
be analytic; at least, so it would seem. Leibniz held (i) that every
proposition has a subject and a predicate; (2) that a substance
is defined by the total of its predicates; (3) that the soul is a
substance. It followed that everything that can be truly said of
a given soul consists in mentioning some predicate which is one
of those that constitute the given soul. “Caesar”, for example, was
a collection of predicates, one of which was “crossing the Rubi-
con”. He was therefore compelled by logic to cross the Rubicon,
and there is no such thing as contingency or free will. Leibniz
ought, on this point, to have agreed with Spinoza, but he chose
not to, for reasons discreditable either to his intellect or to his
moral character. The question is: Can I avoid agreeing with
Spinoza without equal discredit ?

What we have to consider is a subject-predicate proposition
expressing a judgment of perception, such as “this is red”. What

3i7

human knowledge: its scope and limits

is “this”? Clearly it is not my whole momentary experience; I
am not saying “one of the qualities that I am at present experienc-
ing is “redness”. The word “this” may be accompanied by a
gesture, indicating that I mean what is in a certain direction, say
the centre of my visual field. In that case, the core of what I am
saying may be expressed by “centrality and redness overlap
spatially in my present visual field”. It is to be observed that,
within the large complex of my total momentary experience, there
are smaller complexes constituted by spatial compresence in
perceptual space. Whatever quality I see in a certain direction has
perceptual-spatial compresence with the visual quality con-
stituting that direction. It would seem that the word “this”,
accompanied by a gesture, is equivalent to a description, e.g.
“what is occupying the centre of my visual field”. To say that
this description applies to redness is to say something which
clearly is not analytic. But since it employs a description instead
of a name, it is not quite what we set out to consider.

We were considering what sort of thing could have the formal
properties that are required for space-time order. Such a thing
must happen in only one time and place ; it must not recur, either
on another occasion or in another location. So far as time and
physical space are concerned, these conditions are satisfied by
the “complete complex of compresence”, whether this consists
of my momentary experiences or of a full group of overlapping
physical qualities. (I call such a group “full” when, if anything
is added, the members will no longer be all compresent.) But
when we come to consider perceptual space, we have no need
of an analogous procedure. If I see simultaneously two patches
of a given shade of colour, they differ as regards the qualities of
up-and-down, right-and-left, and it is by means of these qualities
that the patches acquire particularity.

With these preliminaries, let us examine the question of proper
names.

It seems preposterous to maintain that “Caesar crossed the
Rubicon” is an analytic proposition. But if it is not, what do we
mean by “Caesar”?

Taking Caesar as he was, without the limitations due to our
ignorance, we may say that he was a series of events, each event
being a momentary total experience. If we were to define “Caesar”
by enumerating these events, the crossing of the Rubicon would

THE PRINCIPLE OF INDIVIDUATION

have to come in our list, and ‘ 4 Caesar crossed the Rubicon” would
be analytic. But in fact we do not define “Caesar” in this way, and
we cannot do so, since we do not know all his experiences. What
happens in fact is more like this: Certain series of experiences
have certain characteristics which make us call such a series a
“person”. Every person has a number of characteristics that are
peculiar to him; Caesar, for example, had the name “Julius Caesar”.
Suppose P is some property which has belonged to only one
person; then we can say: “I give the name ‘A* to the person who
had the property P”. In this case, the name “A” is an abbreviation
for “the person who had the property P”. It is obvious that, if
this person also had the property Q, the statement “A had the
property Q” is not analytic unless Q is analytically a consequence
of P.

This is all very well as regards a historical character, but how
about somebody whom I know more intimately, e.g. myself? How
about such a statement as “I am hot”? This may, following our
earlier analysis, be translated into “heat is one of the qualities
that make up I-now”. Here “I-now” may be taken as denoting
the same complex that is denoted by “my total present momen-
tary experience”. But the question remains: how do I know what
is denoted by “I-now”? What is denoted is continually changing;
on no two occasions can the denotation be the same. But clearly
the words “I-now” have in some sense a constant meaning; they
are fixed elements in the language. We cannot say that, in the
ordinary sense, “I-now” is a name, like “Julius Caesar”, because
to know what it denotes we must know when and by whom it is
used. Nor has it any definable conceptual content, for that,
equally, would not vary with each occasion when the phrase is
used. Exactly the same problems arise in regard to the word
“this”.

But although “I-now” and “this” are not names in quite the
ordinary sense, I incline to think that there is a sense in which
they must count as names. A proper name, as opposed to a
concealed description, can be given to the whole or to any part
of what the speaker is at the moment experiencing. When our
verbal inventiveness fails, we fall back on “this” for the part of
our total momentary experience to which we are specially,attending,
and upon “I-now” for the total momentary experience. I maintain
that I can perceive a complex of compresent qualities without

319

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

necessarily perceiving all the constituent qualities. I can give the
name “this” to such a complex, and then, by attention, observe
that redness (say) is one of its constituent qualities. The resulting
knowledge I express in the sentence “this is red”, which, accord-
ingly, is a judgment of analysis, but not, in the logical sense, an
analytic judgment. A complex can be perceived without my being
aware of all its parts; when, by attention, I become aware that
it has such-and-such a part, this is a judgment of perception
which analyses the whole, but is not analytic, because the whole
was defined as “this”, not as a complex of known parts.

The kind of thing I have in mind is the kind of thing that is
emphasized by the Gestalt psychologists. Suppose I possessed
a clock which showed not only hours and minutes, but the day
of the month, the month of the year, and the year of the Christian
era, and suppose that this clock were to function throughout my
life. It would then never twice during my life present the same
appearance. I might perceive that two appearances of it were
different, without being able to say at once in what the difference
consisted. Attention might lead me to say: “In this appearance
the minute-hand is at the top ; in that, it is at the bottom”. Here
“this” and “that” are merely names, and therefore nothing said
about them can be logically analytic.

There is another way of escaping from the conclusion that
judgments are analytic when in fact they are obviously empirical.
Consider again our clock that never repeats itself. We can define
a date unambiguously by means of this clock. Suppose that, when
the clock indicates io hours 47 minutes on June 15, 1947, I say
“I am hot”. This can be translated into: “Hotness is compresent
with the appearance of the clock that is described as 10 hours
47 minutes, June 15, 1947”. This is certainly not analytic.

One way of making clear the scope and purport of our discussion
is to put it in terms of “minimum vocabularies”. We may ask:
“What is, in principle, a minimum vocabulary for describing the
world of my sensible experience?” We have to ask ourselves:
Can I be content with names of qualities, and words for corn-
presence and for spatial and temporal relations, or do I need also
proper names? And in the latter case, what sorts of things will
need proper names ?

I have suggested that ordinary proper names, such as “Socrates”,
“France”, or “the sun”, apply to continuous portions of space-

320

THE PRINCIPLE OF INDIVIDUATION

time which happen to interest us, and that space-time is composed
of “complete complexes of compresence”, which themselves are
composed of qualities. According to this theory, an “instance”
of (say) a shade of colour is a complex of which that shade is
a constituent. The colour itself exists wherever (as we should
commonly say) there is something that has that colour. Any
collection of compresent qualities may be called a “complex of
compresence”, but it is only a “complete complex” when it cannot
be enlarged without ceasing to be a complex of compresence. Often
a complete complex can be rendered definite by mentioning only
some of its components ; e.g. in the above case of the clock, the
complex is determined when we are told what appearance of the
clock belongs to it. This is what makes dating convenient.

Subject-predicate propositions expressing judgments of per-
ception occur in two ways. First: if a complex is rendered deter-
minate when only some of its constituent qualities are assigned,
we may state that this complex also has such-and-such other
qualities; this is illustrated by the statement “I was hot when the
clock said 10 hours 47 minutes”.

Second : I may perceive a complex without being aware of all
its parts; in that case, I may, by attention, arrive at a judgment
of perception of the form “P is part of W”, where “W” is the
proper name of the perceived complex. If such judgments are
admitted as irreducible, we need proper names for complexes.
But it would seem that the need for such judgments only arises
through ignorance, and that, with better knowledge, our whole
W can always be described by means of its constituents. I think,
therefore, though with some hesitation, that there is no theoretical
need for proper names as opposed to names of qualities and of
relations. Whatever is dated and located is complex, and the
notion of simple “particulars” is a mistake.

As the subject of this Chapter is somewhat difficult, it will
perhaps contribute to clarity and to the prevention of misunder-
standing to repeat the main points of the above discussion more
briefly and less controversially. Let us begin with “compresence”.

“Compresence”, as I wish to understand the term, applies to
the physical world as well as to the world of mind. In the physical
world it is equivalent to “overlapping in space-time”, but this
cannot be taken as its definition, since compresence is needed
in defining spatio-temporal position. I wish to emphasize that the

321

HUMAN knowledge: its scope and limits

relation is to be the very same in physics as in psychology. Just
as many things happen simultaneously in my mind, so, we must
suppose, many things happen simultaneously in every place in
space-time. When we look at the night sky, each star that we can
see produces its separate effect, and this is only possible if, at the
surface of the eye, things are happening that are connected with
each visible star. These different things are all “compresent”.

Wherever several things are compresent, they form what I shall
call a “complex of compresence”. If there are other things com-
present with all of them, they can be added to form a larger
complex. When it is no longer possible to find anything com-
present with all the constituents of the complex, I call the complex
“complete”. Thus a “complete complex of compresence” is one
whose constituents have the two properties {a) that all of them
are compresent, (b) that nothing outside the group is compresent
with every member of the group.

“I-now” denotes the complete complex of compresence which
contains the present contents of my mind. “This” denotes what-
ever part of this complex I am specially noticing.

Complete complexes of compresence are the subjects of spatio-
temporal relations in physical space-time. For empirical, not
logical, reasons, it is highly probable that none of them recurs,
i.e. that none of them precedes itself, or is north of itself, or west
of itself, or above itself.

A complete complex of compresence counts as a space-time
point-instant.

A complex which is not complete will, in general, be a part
of various complete complexes; so will a single quality. A given
shade of colour, for example, is part of every complete complex
which is a space-time point at which this shade exists. To say
of a quality or of an incomplete complex that it “exists at” such-
and-such a space-time point is to say that it is part of the complete
complex which is that point.

An incomplete complex occupies a continuous region in space-
time if, given any two space-time points of which it is part, there
is a continuous route, from the one to the other, consisting wholly
of points of which the incomplete complex is part.

Such a complex may be called an “event”. It has the property
of non-recurrence, but not that of occupying only one space-time
point.

322

THE PRINCIPLE OF INDIVIDUATION

The occupation of a continuous region by a given incomplete
complex may be defined as follows. A complete complex B is said
to be “between” two not too distant complete complexes A and C
if what is common to A and C is part of B. A collection of complete
complexes is “continuous” (for our purposes) if between any two
of its members there are other members of the collection. This,
however, is only a rough-and-ready definition; a precise definition
could only be given by means of topology.

We can never know that a given complex of compresence is
complete, since there may always be something else, of which
we are not aware, which is compresent with every part of the
given complex. This is another way of saying that we cannot,
in practice, define a place or a date exactly.

Certain incomplete complexes have advantages from the point
of view of dating. Take, for example, the date in to-day’s news-
paper together with a 24-hour clock which is going. These two
together make a complex which never recurs, and of which the
duration is so brief that for most purposes we do not need to
notice that it is more than an instant. It is by means of such
incomplete complexes that we in fact determine dates.

For determining spatial position, there are similar advantages
in the ocular qualities of centrality, up-and-down, and right-and-
left. These qualities are mutually exclusive as regards what may
be called “private compresence”, which is a relation between
elements in one total momentary experience. The quality of
centrality, for example, has “private compresence” with the colour
which is occupying the centre of my visual field. The correlation
of places in my private space with places in physical space pro-
ceeds on the assumption that, if visual percepts are not privately
compresent, the corresponding physical objects are not publicly
compresent, but if the visual percepts are privately compresent,
the corresponding physical objects may differ in distance from
the percipient, though they will agree approximately in direction.
Thus private compresence of percepts is a necessary but not
sufficient condition for public compresence of the corresponding
physical objects.

It is to be observed that, in general, every increase in the number
of qualities combined in a complex of compresence diminishes
the amount of space- time that it occupies. A complete complex
of compresence will occupy a portion of space-time which has

323

human knowledge: its scope and limits

no parts that are portions of space-time ; if we assume continuity,
such a portion will have the properties that we expect of a point-
instant. But there is no reason, either empirical or a priori , to
suppose either that space-time is continuous, or that it is not;
everything known can be explained equally well on either hypo-
thesis. If it is not continuous, a finite number of complexes of
compresence will occupy a finite space-time volume, and the
structure of space-time will be granular, like that of a heap of
shot.

A complex of compresence, as I conceive it, is determinate
when the qualities constituting it are given. That is to say, if the
qualities q l9 q 2> . . . q Q are all mutually compresent, there is just
one complex of compresence, say C, which consists of the com-
bination of these qualities. It is always logically possible for C
to occur more than once, but I assume that, if C is sufficiently
complex, there will not in fact be recurrence. A few words are
necessary to explain what, logically, is meant by “recurrence”.
Let us, for simplicity, confine ourselves to time in one biography,
and let us begin by considering complete complexes.

I assume that, between any two complete complexes belonging
to the same biography, there is a relation of earlier-and-later.
To suppose that a complete complex can recur is to suppose that
a complete complex can have the relation of earlier-and-later to
itself. This, I assume, does not happen, or at any rate does not
happen within any ordinary period of time. I do not mean to
deny dogmatically that history may be cyclic, as some Stoics
thought, but the possibility is too remote to need to be taken
into account.

Since we can never know that a known complex of compresence
is complete — since, in fact, we can be pretty sure that it is not —
we use, in practice, for purposes of chronology and geography,
such incomplete complexes as either do not recur at all, or recur
in a fairly regular manner. The date on a calendar persists for
twenty-four hours, and then changes abruptly. Some clocks have
a minute-hand that moves with a jerk once a minute; the appear-
ance of such a clock persists for a minute and recurs every twelve
hours. If we had sixty such clocks in a circle, and each gave its
jerk one second after the one to the left of it, the complex con-
sisting of the appearance of all the sixty would fix the time within
one second. By such methods accuracy of dating may be indefi-

3*4

THE PRINCIPLE OF INDIVIDUATION

nitely increased. Exactly similar remarks apply to methods of
determining latitude and longitude.

A complex of compresence, though defined when all its con-
stituent qualities are given, is not to be conceived, like a class,
as a mere logical construction, but as something which can be
known and named without our having to know all its constituent
qualities. The logical point involved may be made clear as follows:
the relation earlier-and-later holds, primarily, between two com-
plete complexes of compresence, and only in a derivative and
definable sense between partial complexes. In the case of a purely
logical structure, a statement about the structure can be reduced
to one about its components, but in the case of the time-order
this is not possible on the theory of “particulars” adopted in this
Chapter. A complex can, therefore, be mentioned in a way which
is not reducible to a statement about any or all of its constituents.
It is, in fact, the sort of object that is a “this”, and that can have
a proper name. A given collection of qualities only forms a com-
plex of compresence if the qualities happen to be all mutually
compresent; when they are, the complex is something new, over
and above the qualities, though necessarily unique when the
qualities are given.

According to the above theory, a complex of compresence which
does not recur takes the place traditionally occupied by “particu-
lars”; a single such complex, or a string of such complexes
causally connected in a certain way, is the kind of object to which
it is conventionally appropriate to give a proper name. But a
complex of compresence is of the same logical type as a single
quality, that is to say, any statement which is significant about
either is significant, though probably not true, about the other.

We may agree with Leibniz to this extent, that only our
ignorance makes names for complexes necessary. In theory, every
complex of compresence can be defined by enumerating its com-
ponent qualities. But in fact we can perceive a complex without
perceiving all its component qualities ; in this case, if we discover
that a certain quality is a component of it, we need a name for
the complex to express what it is that we have discovered. The
need for proper names, therefore, is bound up with our way of
acquiring knowledge, and would cease if our knowledge were
completer

3 2 S

Chapter IX
CAUSAL LAWS

T HE practical utility of science depends upon its ability to
foretell the future. When the atomic bombs were dropped,
it was expected that large numbers of Japanese would die,
and they did. Such highly satisfactory results have led, in our day,
to an admiration of science, which is due to the pleasure we derive
from the satisfaction of our lust for power. The most powerful
communities are the most scientific, though it is not the men of
science who wield the power conferred by their knowledge. On
the contrary, the actual men of science are rapidly sinking into the
position of state prisoners, condemned to slave labour by brutal
masters, like subject djinns in the Arabian Nights. But we must
not waste any more time upon such pleasant topics. The power of
science is due to its discovery of causal laws, and it is causal laws
that are to occupy us in this chapter.

|^A “causal law”, as I shall use the term, may be defined as a
general principle in virtue of which, given sufficient data about
certain regions of space-time, it is possible to infer something
about certain other regions of space-time. The inference may be
only probable, but the probability must be considerably more than
a half if the principle in question is to be considered worthy to be
called a “causal law”.

I have purposely made the above definition very wide. In the
first place, the region to which we infer need not be later than those
from which we infer. There are, it is true, some laws — notably
the second law of thermodynamics — which allow inferences for-
wards more readily than backwards, but this is not a general
characteristic of causal laws. In geology, for example, the inferences
are almost all backwards. In the second place, we cannot lay
down rules as to the number of data that may be involved in
stating a law. If it should ever become possible to state the laws
of embryology in terms of physics, enormously complex data
would be required. In the third place, the inference may be only
to some more or less general characteristic of the inferred event
or events. In the days before Galileo it was known that un-
supported heavy bodies fall, which was a causal law; but it was

326

CAUSAL LAWS

not known how fast they fall, so that when a weight was dropped
it was impossible to say accurately where it would be after a given
lapse of time. In the fourth place, if the law states a high degree
of probability it may be almost as satisfactory as if it stated a
certainty. I am not thinking of the probability of the law being
true ; causal laws, like the rest of our knowledge, may be mistaken.
What I am thinking of is that some laws state probabilities, for
example the statistical laws of quantum theory. Such laws,
supposing them completely true, make inferred events only
probable, but this does not prevent them from counting as causal
laws according to the above definition.

One advantage of admitting laws which only confer probability
is that it enables us to incorporate in science the crude generaliza-
tions from which common sense starts, such as “fire burns”,
“bread nourishes”, “dogs bark”, or “lions are fierce”. All these
are causal laws, and all are liable to exceptions, so that in a given
case they confer only probability. The fire on a plum pudding
does not burn you, poisoned bread does not nourish, some dogs
are too lazy to bark, and some lions grow so fond of their keepers
that they cease to be fierce. But in the great majority of cases the
above generalizations will be a sound guide in action. There are
a large number of such approximate regularities which are assumed
in our every-day behaviour, and it is from them that the conception
of causal laws arose. Scientific laws, it is true, are no longer so
simple: they have become complicated in the endeavour to give
them a form in which they are not liable to exceptions. But the
old simpler laws remain valid so long as they are only regarded
as asserting probabilities.

Causal laws are of two sorts, those concerned with persistence
and those concerned with change. The former kind are often not
regarded as causal, but this is a mistake. A good example of a law
of persistence is the first law of motion. Another example is the
persistence of matter. After the discovery of oxygen, when the
process of combustion came to be understood, it was possible to
regard all matter as indestructible. It has now become doubtful
whether this is quite true, but it remains true for most practical
purposes. What appears to be more exactly true is the persistence
of energy. The gradual development of laws stating persistence
started from the common- sense belief, based on pre-scientific
experience, that most solid objects continue to exist until they

327

human knowledge: its scope and limits

crumble from old age or are destroyed by fire, and that, when this
happens, it is possible to suppose that their small parts survive
in a new arrangement. It was this pre-scientific point of view that
gave rise to the belief in material substance.

Causal laws concerned with change were found by Galileo and
Newton to demand statement in terms of acceleration, i.e. change
of velocity in magnitude or direction or both. The greatest
triumph of this point of view was the law of gravitation, according
to which every particle of matter causes in every other an accelera-
tion directly proportional to the mass of the attracting particle
and inversely proportional to the square of the distance between
them. But Einstein's form of the law of gravitation made it more
analogous to the law of inertia, and, in a sense, a law of persistence
rather than a law of change. According to Einstein, space-time is
full of what we may call hills ; each hill grows steeper as you go
up, and has a piece of matter at the top. The result is that the
easiest route from place to place is one which winds round the hills.
The law of gravitation consists in the fact that bodies always take
the easiest route, which is what is called a “geodesic". There is a
law of cosmic laziness called the “principle of least action", which
states that when a body moves from one place to another it will
choose the route involving least work. By means of this principle
gravitation is absorbed into the geometry of space-time.

The essential laws of change in modern physics are those of
quantum theory, which govern transitions from one form of
energy to another. An atom can emit energy in the form of light,
which then travels on unchanged until it meets another atom,
which may absorb the energy of the light. Such interchanges are
governed by certain rules, which do not suffice to say what will
happen on a given occasion, but can predict, with a very high
degree of probability, the statistical distribution of possible
happenings among a very large number of interchanges. This is
as near as physics can get at present to the ultimate character of
causal laws.

Everything that we believe ourselves to know about the physical
world depends entirely upon the assumption that there are
causal laws. Sensations, and what we optimistically call “percep-
tions", are events in us. We do not actually see physical objects,
any more than we hear electromagnetic waves when we listen to
the wireless. What we directly experience might be all that exists,

328

CAUSAL LAWS

if we did not have reason to believe that our sensations have
external causes. It is important, therefore, to inquire into our
belief in causation. Is it mere superstitition, or has it a solid
foundation ?

The question of the justification of our belief in causality be-
longs to theory of knowledge, and I shall therefore postpone it
for the present. My purpose in this Part is the interpretation of
science, not an inquiry into the grounds for supposing science
valid. Science assumes causality in some sense, and our present
question is: in what sense is causality involved in scientific
method P 1

Broadly speaking, scientific method consists in inventing
hypotheses which fit the data, which are as simple as is compatible
with this requirement, and which make it possible to draw infer-
ences subsequently confirmed by observation. The theory of
probability shows that the validity of this process depends upon
an assumption which may be roughly stated as the postulate that
there are general laws of certain kinds. This postulate, in a
suitable form, can make scientific laws probable, but without it
they do not even achieve probability. We have therefore to
examine this assumption, to find out the most plausible form in
which it is both effective and possibly true.

If there is no limit to the complexity of possible laws, every
imaginable course of events will be subject to laws, and therefore
the assumption that there are laws will become a tautology. Take,
for example, the numbers of all the taxis that I have hired in the
course of my life, and the times when I have hired them. We have
here a finite set of integers and a finite number of corresponding
times. If n is the number of the taxi that I hired at the time t y it
is certainly possible, in an infinite number of ways, to find a
function / such that the formula

n =f(t)

is true for all the values of n and t that have hitherto occurred.
An infinite number of these formulae will fail for the next taxi
that I hire, but there will still be an infinite number that remain
true. By the time I die, it will be possible to close the account, and
there will still remain an infinite number of possible formulae,

1 The following pages anticipate, in an abbreviated form, the fuller
discussions of Parts V and VI.

329

HUMAN knowledge: its scope and limits

each of which might claim to be a law connecting the number of
a taxi with the time when I hire it.

The merit of this example, for my present purpose, is its
obvious absurdity. In the sense in which we believe in natural
laws we should say that there is no law connecting the n and / of
the above formula, and that, if any suggested formula happens to
work, that is a mere chance. If we had found a formula that
worked in all cases up to the present, we should not expect it to
work in the next case. Only a superstitious person whose emotions
are involved will believe an induction of this sort; gamblers at
Monte Carlo practise inductions which no man of science would
sanction. But it is not altogether easy to state the difference be-
tween the inductions of the superstitious gambler and the in-
ductions of the prudent man of science. Obviously there is a
difference, but in what does it consist ? And is the difference such
as to affect logical validity, or does it consist merely in a difference
as to the obviousness of the appeal to the emotions ? Is the faith in
scientific method merely the scientist’s superstition appropriate
to his kind of gambling? These questions, however, belong to the
theory of knowledge. For the present I want to discover not why
we believe, but what we believe, when we believe in natural
laws.

It is customary to speak of induction as what is needed to make
the truth of scientific laws probable. I do not think that induction,
pure and simple, is fundamental. The above example of the
numbers of taxis illustrates this. All past observations as to these
numbers are compatible with a number of laws of the form
n — f (t), and these will, as a rule, give different values for the
next n. We cannot therefore use them all for prediction, and in
fact we have no inclination to believe in any of them. Generalizing,
we may say: Every finite set of observations is compatible with a
number of mutually inconsistent laws, all of which have exactly
the same inductive evidence in their favour. Therefore pure
induction is invalid, and is, moreover, not what we in fact believe.

Whenever inductive evidence seems to us to make a suggested
law very probable, the law is one which had suggested itself more
or less independently of the evidence, and had seemed to us in
some way likely to be true. When this is the case, subsequent
confirmatory evidence is found astonishingly convincing.

This, however, is only partially true. If a law is suggested of

330

CAUSAL LAWS

which the consequences are very different from what we should
expect, and it then is confirmed by observation, we are more prone
to believe in it than if its results were commonplace. But in such
a case the law itself may seem plausible, although its consequences,
when mentioned, are found surprising. Perhaps one of the most
important effects of scientific education is to modify the hypo-
theses that appear prima facie probable. It was this cause, not direct
negative evidence, that led the belief in witchcraft to decay. If you
had a number of outwardly similar boxes, of which some contained
gyrostats, and you showed them to a savage, saying that by
uttering a magic formula you could make any one of them im-
possible to turn round, the inductive evidence would soon per-
suade him that you were right, but a man of educated scientific
outlook would search for some other explanation in spite of
repeated apparent verifications of your “law”.

Induction, moreover, does not validate many of the inferences
in which science feels most confidence. We are all convinced that,
when a number of people hear a sound simultaneously, their
common experience has an external source, which is propagated
through the intervening medium by sound waves. There cannot
be inductive evidence (unless in some extended sense) for some-
thing outside human experience, such as a sound wave. Our
experience will be the same whether there really are sound-waves,
or, though there are none, auditory sensations occur as they
would if there were sound-waves ; no inductive evidence can ever
favour one of these hypotheses rather than the other. Nevertheless
every one in fact accepts the realist alternative — even the idealist
philosopher except in his professional moments. We do this on
grounds that have nothing to do with induction — partly because
we like laws to be as simple as possible, partly because we believe
that causal laws must have spatio-temporal continuity, i.e. must
not involve action at a distance.

In the establishment of scientific laws experience plays a two-
fold part. There is the obvious confirming or confuting of a hypo-
thesis by observing whether its calculated consequences take
place, and there is the previous experience which determines
what hypotheses we shall think antecedently probable. But behind
these influences of experience there are certain vague general
expectations, and unless these confer a finite a priori probability
on certain kinds of hypotheses, scientific inferences are not valid.

33i

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

In clarifying scientific method it is essential to give as much
precision as possible to these expectations, and to examine whether
the success of science in any degree confirms their validity. After
being made precise the expectations are, of course, no longer
quite what they were while they remained vague, but so long as
they remain vague the question whether they are true or false is
also vague.

It seems to me that what may be called the “faith” of science
is more or less of the following sort: there are formulae (causal
laws) connecting events, both perceived and unperceived; these
formulae exhibit spatio-temporal continuity, i.e. involve no direct
unmediated relation between events at a finite distance from each
other; a suggested formula having the above characteristics be-
comes highly probable if, in addition to fitting in with all past
observations, it enables us to predict others which are subse-
quently confirmed and which would be very improbable if the
formula were false.

The justification of this “faith”, if any, belongs to theory of
knowledge. Our present task is completed in having stated it. But
there is still need of some discussion as to the origin and growth
of this “faith”.

There are various possible postulates which can be taken as
the basis of scientific method, but it is difficult to state them with
the necessary precision. There is the law of causality; there is the
uniformity of nature ; there is the reign of law : there is the belief
in natural kinds, and Keynes’s principle of limited variety; and
there is structural constancy with spatio-temporal continuity. It
ought to be possible, out of all these somewhat vague assumptions,
to distil some definite axiom or axioms which, if true, will confer
the desired degree of probability on scientific inferences.

The principle of causality appears in the works of almost all
philosophers in an elementary form which it never takes in any
advanced science. They suppose science to assume that, given any
suitable class of events A, there is always some other class of
events B such that every A is “caused” by a B; moreover every
event belongs to some such class.

Most philosophers have held that “cause” means something
different from “invariable antecedent”. The difference may be
illustrated by Geulincx’s two clocks, which both keep perfect
time ; when one points to the hour, the other strikes, but we do not

332

CAUSAL LAWS

think that the one has “caused” the other to strike. A non-
scientific Fellow of my College lately remarked in despair: “The
barometer has ceased to have any effect on the weather”. This was
felt to be a joke, but if “cause” meant “invariable antecedent” it
would not be. It is supposed that when A is caused by B the
sequence is not merely a fact, but is in some sense necessary. This
conception is bound up with the controversy about free will and
determinism, summed up by the poet in the following lines :

There was a young man who said : Damn !

I learn with regret that I am
A creature that moves
In predestinate grooves,

In short, not a bus, but a tram.

As against this view most empiricists have held that “cause”
means nothing but “invariable antecedent”. The difficulty of this
view, and indeed of any suggestion that scientific laws are of the
form “A causes B”, is that such sequences are seldom invariable,
and, even if they are invariable in fact, circumstances can easily
be imagined which would prevent them from being so. As a rule,
if you tell a man he is a silly fool he will be angry, but he may be a
saint, or may happen to die of apoplexy before he has time to lose
his temper. If you strike a match on a box it usually lights, but
sometimes it breaks or is damp. If you throw a stone in the air it
usually falls down again, but it may be swallowed by an eagle
under the impression that it is a bird. If you will to move your
arm it usually moves, but not if you are paralysed. In such ways
all laws of the form “A causes B” are liable to exceptions, since
something may intervene to prevent the expected result.

Nevertheless, there are reasons, of which the strength will
appear in Part VI, for admitting laws of the form “A causes B”,
provided that we do so with suitable safeguards and limitations.
The concept of more or less permanent physical objects, in its
common-sense form, involves “substance”, and when “substance”
is rejected we have to find some other way of defining the identity
of a physical object at different times. I think this must be done
by means of the concept “causal line”. I call a series of events a
“causal line”, if given some of them, we can infer something about
the others without having to know anything about the environment.
For example, if my doors and windows are shut, and at intervals

333

HUMAN knowledge: its scope and limits

I notice my dog asleep on the hearthrug, I infer that he was there,
or at least somewhere in the room, at the times when I was not
noticing him. A photon which travels from a star to my eye is a
series of events obeying an intrinsic law, but ceasing to obey this
law when it reaches my eye. When two events belong to one
causal line, the earlier may be said to “cause” the later. In this
way laws of the form 44 A causes B” may preserve a certain
validity. They are important in connection both with perception
and with persistence of material objects.

It is the possibility of something intervening that has led
physics to state its laws in the form of differential equations,
which may be regarded as stating what is tending to happen. And
as already explained, classical physics, when presented with
several causes acting simultaneously, represents the resultant as
a vector sum, so that, in a sense, each cause produces its effect as
if no other cause were acting. But in fact the whole conception
of “cause” is resolved into that of “law”. And laws, as they occur
in classical physics, are concerned with tendencies at an instant.
What actually happens is to be inferred by taking the vector sum
of all the tendencies at an instant, and then integrating to find
out the result after a finite time.

All empirical laws are inferred from a finite number of observa-
tions, eked out by interpolation and extrapolation. The part
played by interpolation is not always adequately realized. Take
for example the apparent motions of the planets. We assume that,
during the day-time, they pursue a smooth course which fits in
easily with their observed courses during the preceding and
succeeding nights. It would be a possible hypothesis that planets
only exist when they are observed, but this would make the laws
of astronomy very complicated. If it is objected that planets can
be photographed fairly continuously, the same problem arises as
regards the photographs : do they exist when no one is looking at
them? This again is a question of interpolation, and the interpola-
tion is justified by the fact that it gives the simplest laws com-
patible with what has been observed.

Exactly the same principle applies to extrapolation. Astronomy
makes assertions, not only about what planets have done at all
times since there were astronomers, but about what they will do
and what they did before there was any one to notice them. This
extrapolation is often spoken of as if it involved some principle

334

CAUSAL LAWS

other than that involved in interpolation, but in fact the principle
is one and the same : to choose the simplest law that fits the known
facts.

As a postulate, however, this is open to grave objections.
“Simple” is a vague conception. Moreover it often happens that
a simple law turns out, after a time, to be too simple, and that the
correct law is more complicated. But in such cases the simple law
is usually approximately right. If, therefore, we only assert that
a law is approximately right, we cannot be convicted of error
when some other law is found to be a still better approximation.

The uniformity of nature, which is a principle sometimes
invoked, has no definite meaning except in connection with
natural laws. If it is already granted that there are natural laws,
the principle of the uniformity of nature states that time and place
must not appear explicitly in the formulation of laws: the laws
must be the same in one part of space- time as in another. This
principle may or may not be true, but in any case it is insufficient
as a postulate, since it presupposes the existence of laws.

The existence of natural kinds underlies most pre-scientific
generalizations, such as “dogs bark” or “wood floats”. The
essence of a “natural kind” is that it is a class of objects all of
which possess a number of properties that are not known to be
logically interconnected. Dogs bark and growl and wag their
tails, while cats mew and purr and lick themselves. We do not
know why all the members of an animal species should share so
many common qualities, but we observe that they do, and base
our expectations on what we observe. We should be amazed if
a cat began to bark.

Natural kinds are not only of biological importance. Atoms and
molecules are natural kinds; so are electrons, positrons, and
neutrons. Quantum theory has introduced a new form of natural
kinds in its discrete series of energy levels. It is now possible to
conceive the ultimate structure of the physical world not as a
continuous flux, in the manner of conventional hydrodynamics,
but in a more Pythagorean fashion, in which models are derived
from analogy with a heap of shot. Evolution, which in Darwin’s
time “broadened slowly down from precedent to precedent”,
now takes revolutionary leaps by means of mutants, or freaks.
Perhaps wars and revolutions have made us impatient of gradual-
ness; however that may be, modem scientific theories are much

335

HUMAN knowledge: its scope and limits

more jolty and jagged than the smooth cosmic stream of ordered
progress imagined by the Victorians.

The bearing of all this on induction is of considerable importance.
If you are dealing with a property which is likely to be character-
istic of a natural kind, you can generalize fairly safely after very
few instances. Do seals bark? After hearing half a dozen do so,
you confidently answer “yes”, because you are persuaded in
advance that either all seals bark or no seals bark. When you
have found that a few pieces of copper are good conductors of
electricity, you unhesitatingly assume that this is true of all
copper. In such cases a generalization has a finite a priori proba-
bility, and induction is less precarious than in other problems.

Keynes has a postulate by which, in his opinion, inductive
arguments might be justified ; he calls it the principle of limited
variety. It is a form of the assumption of natural kinds. This is
one of the expedients in the way of a general assumption which,
if true, validates scientific method. I shall have more to say about
it at a later stage. What has been said in this chapter is only by
way of anticipation.

336

Chapter X

SPACE-TIME AND CAUSALITY

P hysical events are arranged by physics in a four-dimen-
sional manifold called space-time. This manifold is an
improvement on the older manifold of “things” arranged
in varying spatial patterns at varying times ; and this, in turn, was
an improvement upon the manifold resulting from assuming an
accurate correspondence between percepts and “things”. No
doubt physics would like to forget its early history, which, like
that of many established institutions, is not so creditable as could
be wished. But unfortunately its title to our allegiance is difficult
to disentangle from its early association with naive realism ; even
in its most sophisticated form, it still appears as an emendation,
for which naive realism supplies the text.

Perceptual space is a common- sense construction, composed of
diverse raw materials. There are visual space- relations : up-and-
down, right-and-left, depth up to a certain distance (after which
differences of depth become imperceptible). There are the differ-
ences in sensations of touch which enable us to distinguish a
touch on one part of the body from a touch on another. There is
the somewhat vague power of estimating the direction of a sound.
Then there are experienced correlations, of which the most
important is the correlation of sight and touch ; there are observa-
tions of movement, and the experience of moving parts of our
own body.

Out of such raw materials (the above list does not claim to
be complete) common sense constructs a single space containing
objects perceived and unperceived, the perceived objects being
identified with percepts, according to the principles of naive
realism. The unperceived objects, for common sense, are those
which we should perceive if we were in the right position and
with suitably adjusted sense-organs, together with objects only
perceived by others, and objects, such as the interior of the
earth, which are perceived by no one but inferred by common
sense.

In the passage from the common-sense world to that of physics,
certain common-sense assumptions are retained, though in a

337

HUMAN knowledge: its scope and limits

modified form. For instance, we assume that the furniture of our
room continues to exist when we do not see it. Common sense
supposes that what continues is just what we see when we look,
but physics says that what continues is the external cause of what
we see, i.e. a vast assemblage of atoms undergoing frequent
quantum transformations. In the course of these transformations
they radiate energy, which, when it comes in contact with a
human body, has various effects, some of which are called “per-
ceptions”. Two simultaneous parts of one visual percept have a
certain visual spatial relation which is a component of the total
percept; the physical objects which correspond to these parts of
my total percept have a relation roughly corresponding to this
visual spatial relation. When I say that the relation “corresponds”,
I mean that it is part of a system of relations having, to some
extent, the same geometry as that of visual percepts, and that the
location of physical objects in physical space has discoverable
relations to the location of perceptual objects in perceptual space.

But this correspondence is by no means exact. Let us take, to
simplify our problem, the heavenly bodies as they are and as they
appear. As they appear, they do not obviously differ as regards
distance from us; they look like bright points or patches on the
celestial sphere. That is to say, their position in visual space is
defined by only two co-ordinates. But eclipses and occultations
soon led to the view that they are not in fact all equi-distant from
the earth, though it was a long time before differences of distance
among the fixed stars were admitted. To fix the position of a
heavenly body relatively to ourselves we need three polar co-
ordinates, r, 6 , <f>. It was assumed that 9 and <f> could be the same
for the physical star as for the perceived star, but r must be
computed; in fact a great deal of astronomy has been concerned
with computing r. The assumption that 6 and <f> are the same in
visual and physical space is equivalent to the assumption that
light travels in straight lines. This assumption, after a time, came
to be thought not exactly true, but it is still sufficiently true for
a first approximation.

The 6 and <f> of astronomical space, though they have approxi-
mately the same numerical measure as the 6 and <f> of visual
space, are not identical with the latter. If they were identical, the
hypothesis that light does not move exactly in straight lines would
be meaningless. This illustrates at once the connection and the

33 8

SPACE-TIME AND CAUSALITY

difference between visual space when we look at the night sky
and astronomical space as constructed by the astronomers. The
connection is kept as close as may be, but beyond a point it has to
be abandoned if we are to believe in comparatively simple laws
governing the real and apparent movements of the heavenly
bodies.

Small distances from ourselves are not estimated by the elaborate
methods required in astronomy. We can roughly “see” small
distances, though the stereoscope produces this effect deceptively.
We judge things that touch our body to be close to the part they
touch. When things are not touching us, we sometimes can move
so as to come in contact with them; the amount of movement
required measures, roughly, their initial distance from us. We
have thus three common-sense ways of estimating the distance of
visual objects on the surface of the earth. Scientific ways of
estimating distance use these ways as their foundation, but
correct them by means of physical laws inferred by assuming them.
The whole process is one of tinkering. If common-sense estimates
of distances and sizes are roughly correct, then certain physical
laws are roughly correct. If these laws are quite correct, the
common-sense estimates must be slightly amended. If the various
laws are not exactly compatible, they must be adjusted until the
inconsistency ceases. Thus observation and theory interact;
what, in scientific physics, is called an observation is usually
something involving a considerable admixture of theory.

Let us now abandon the consideration of the stages towards
theoretical physics, and compare the finished physical world with
the world of common sense. I see, let us suppose, a buttercup and
a bluebell; common sense says the buttercup is yellow and the
bluebell is blue. Physics says that electromagnetic waves of many
different frequencies start from the sun and reach the two flowers ;
when they reach them, the buttercup scatters the waves whose
frequency produces a yellow sensation, and the bluebell those
that produce a blue sensation. This difference in the effect of the
two flowers is assumed to be due to some difference in their
structure. Thus although yellow and blue exist only where there
is an eye, the difference between them allows us to infer differences
between the physical objects in the directions in which we see
yellow and blue respectively.

Common sense constructs a single space containing “things”

339

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

which combine properties revealed by different senses, such as hot
and hard and bright. These “things” are placed by common
sense in a three-dimensional space, in which distance cannot be
estimated by common-sense methods unless it is small. Physics
until recently retained something like “things”, but called it
“matter”, and robbed it of all properties except position in space.
The position of a piece of matter in space was roughly identical
with that of the corresponding “thing”, except that the distance,
if great, had to be calculated by rather elaborate scientific methods.

In this picking and choosing among common-sense beliefs,
physics has acted without formulated principles, but nevertheless
on a subconscious plan which we must try to make explicit. Part
of this plan is to retain always as much of the common-sense
world as is possible without intolerable complication; another
part is to make such non-refutable assumptions as will lead to
simple causal laws. This latter procedure is already implicit in
the common-sense belief in “things”: we do not believe that the
visible world ceases to exist when we shut our eyes, and we hold
that the cat exists when it is secretly stealing the cream as well as
when we are punishing it for doing so. All this is “probable”
inference: it is logically possible to suppose that the world consists
only of my percepts, and the inference to the common-sense
world, as to that of physics, is non- demonstrative. But I do not
wish to go behind common sense at present; I wish only to
consider the transition from common sense to physics.

Modern physics is further from common sense than the physics
of the nineteenth century. It has dispensed with matter, substi-
tuting series of events ; it has abandoned continuity in microscopic
phenomena; and it has substituted statistical averages for strict
deterministic causality affecting each individual occurrence. But
it has still retained a great deal of which the source is common
sense. And there are still continuity and determinism so far as
macroscopic phenomena are concerned, and for most purposes
there is still matter.

The world of physics contains more than the world of percepts,
and in some respects contains more than the world of common
sense. But while it exceeds both in quantity, it falls short of both
in known qualitative variety. Both common sense and physics
supplement percepts by the assumption that things do not cease
to exist when unperceived, and by the further assumption that

340

SPACE-TIME AND CAUSALITY

things never perceived can often be inferred. Physics supplements
the common-sense world by the whole theory of microscopic
phenomena; what it asserts about atoms and their history sur-
passes what common sense allows itself to infer.

There are two specially important kinds of chains of events:
first, those which constitute the history of a given piece of matter;
second, those which connect an object with the perception of it.
The sun, for instance, has a biography consisting of all that happens
in the part of space-time that it occupies ; this biography may be
said to be the sun. It also emits radiations, some of which reach
eyes and brains, and cause the sort of occurrence which is called
“seeing the sun”. Broadly speaking, the former set of events
consists of quantum transitions, the latter of radiant energy.
There are correspondingly two sets of causal laws, one set con-
necting events belonging to the same piece of matter, the other
connecting parts of the same radiation. There is also a third set of
laws, concerning the transition from energy in the atom to radiant
energy and vice versa.

Perceiving, as we know it introspectively, appears to be some-
thing quite different from the events that physics considers. There-
fore if there is to be inference from percepts to physical occurrences,
or from physical occurrences to percepts, we need laws which,
prima facie , are not physical. I incline to think that physics can
be so interpreted as to include these laws, but for the present I
shall not consider this possibility. Our problem is, therefore:
taking percepts as we know them in experience, and physical
occurrences as asserted by physics, what laws do we know that
inter-connect the two and therefore allow inference from one to
the other ?

In part, the answer is already patent to common sense. We see
when light strikes the eye, we hear when sound strikes the ear, we
have sensations of touch when the body is in contact with some-
thing else, and so on. These laws are not laws of physics or physio-
logy, unless physics is subjected to a radical re- interpretation.
They are laws stating the physical antecedents of perceptions.
These antecedents are partly outside the percipient’s body
(except when he is perceiving something in his own body), partly
in his sense-organs and nerves, partly in his brain. A failure in any
of these antecedents prevents the perception. But conversely,
if one of the later antecedents is caused in an unusual way, the

34i

human knowledge: its scope and limits

percept will be what it would have been if the causation had been
usual, and the percipient is liable to be deceived — for example by
something seen in a mirror or heard on the wireless, if he is un-
accustomed to mirrors and wireless.

Each single inference from a perception to a physical object is
therefore liable to be erroneous in the sense of causing expectations
that are not fulfilled. It will not usually be erroneous in this sense,
since the habit of making that sort of inference must have been
generated by a number of occasions when the inference was
justified. But here a little further precision is necessary. From a
practical point of view, an inference from a percept is justified
if it gives rise to expectations that are verified. This, however, is
all within the realm of percepts. All that strictly follows is that
our inferences as to physical objects are consistent with exper-
ience, but there may be other hypotheses that are equally con-
sistent.

The justification of our inferences from perception to physical
objects depends upon the consistency of the whole system. First,
from ordinary perceptions, we arrive at an elementary kind of
physics; this suffices to cause us to put in a separate category
dreams, mirages, etc., which contradict our elementary physics.
We then set to work to improve our elementary physics so as to
include the exceptional phenomena; there is, for instance, a
perfectly good physical theory of mirages. We learn in this way to
be critical, and we form the concept of a “trained observer”. We
are critical of percepts in the name of laws, and of laws in the
name of percepts; gradually, as physics improves, a closer and
closer harmony between percepts and laws is established.

But when I say that we become critical of percepts, I must
guard against a misunderstanding. Percepts certainly occur, and a
theory which has to deny any of them is faulty ; but some, being
caused in an unusual way, lead common sense into erroneous
inferences. Of this the mirage is a good example. If I see a lake
which is only a mirage, I see what I see just as truly as if there
were a physical lake; I am mistaken, not as to the percept, but
as to what it implies. The percept makes me think that if I walk
in a certain direction, I shall reach water that I can drink, and
in this I am deceived ; but my visual percept may be exactly what
it would be if there really were water. My physics, if adequate,
must explain not only that there is no water, but also why there

342

SPACE-TIME AND CAUSALITY

seems to be water. A mistaken perception is mistaken, not as to
the percept itself, but as to its causal correlates and antecedents
and consequents; frequently the mistake is in an animal inference.
The fact that animal inferences may be mistaken is one reason for
classifying them as inferences.

The relation of physical laws to experience is not altogether
simple. Broadly speaking, laws can be disproved by experience,
but not proved by it. That is to say, they assert more than ex-
perience alone would warrant. In the case of the mirage, if I have
believed it real, and have also assumed that a large lake will not
dry up in a few hours, I can discover that the mirage caused me to
have a false belief. But the false belief may have been the belief
that the lake could not dry up so quickly. The belief in the per-
sistence of material objects throughout the interval between two
occasions when they are observed is one which, as a matter of
logic, cannot be proved by observation. Suppose I were to set up
the hypothesis that tables, whenever no one is looking, turn into
kangaroos ; this would make the laws of physics very complicated,
but no observation could refute it. The laws of physics, in the
form in which we accept them, must not only be in agreement
with observation, but must, as regards what is not observed, have
certain characteristics of simplicity and continuity which are not
empirically demonstrable. In general, we think that physical
phenomena are not affected by being observed, although this is
not thought to be strictly true as regards the minute phenomena
upon which quantum theory is based.

Physics, assuming it perfected, would have two characteristics.
In the first place, it would be able to predict percepts ; no percep-
tion would be contrary to what physics had led us to expect. In
the second place, it would assume unobserved physical occurrences
to be governed by causal laws as similar as possible to those that
we infer from cases of continuous observation. For example, if I
watch a moving body, the motion that I see is sensibly continuous ;
I therefore assume that all motion, whether observed or not, is
approximately continuous.

This brings us to the question of causal laws and physical
space-time.(Physical space-time, as we have seen, is an inference
from perceptual space and time; it contains all observed occur-
rences, and also all unobserved occurrences. But since it is in-
ferential, the location of an occurrence in it is also inferential.

343

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

The locating of events in physical space-time is effected by two
methods. First, there is a correlation between perceptual space
and time and physical space-time, though this correlation is only
rough and approximate. Second, the causal laws of physics assign
an order to the events concerned, and it is partly by means of them
that unobserved events are located in space- time .S

A causal law, as I use the term, is any law which, if true, makes
it possible, given a certain number of events, to infer something
about one or more other events. For example, “planets move in
ellipses” is a causal law. If this law is true, since five points de-
termine an ellipse, five data (theoretically) should enable us to
calculate the orbit of the planet. Most laws, however, have not
this simplicity ; they are usually expressed in differential equations.
When they are so expressed they assume an order: each event
must have four co-ordinates, and neighbouring events are those
whose co-ordinates are very nearly the same. But the question
arises : how do we assign co-ordinates to events in physical space-
time? I maintain that, in doing so, we make use of causal laws.
That is to say, the relation of causal laws to space-time order is a
reciprocal one. The correct statement is: Events can be arranged
in a four-dimensional order such that, when so arranged, they
are interconnected by causal laws which are approximately
continuous, i.e. events whose co-ordinates differ very little also
differ very little. Or rather: Given any event, there is a series of
closely similar events, in which the time- co-ordinate varies
continuously from rather less to rather more than that of the
given event, and in which the space-co-ordinates vary continuously
about those of the given event. This principle, apparently, does
not hold for quantum transitions, but it holds for macroscopic
events, and for all events (such as light-waves) where there is no
matter.

The correlation between physical and perceptual space-time,
which is only approximate, proceeds as follows. In visual space,
if objects are near enough for differences of depth to be per-
ceptible, every visual percept has three polar co-ordinates, which
may be called distance, up-and-downness, and right-and-leftness.
All these are qualities of the percept, and all are measurable. We
may assign the same numerical co-ordinates to the physical
object which we are said to be seeing, but these co-ordinates no
longer have the same meaning as they have in visual space. It is

344

SPACE-TIME AND CAUSALITY

because they do not have the same meaning that it is possible for
the correlation to be only rough — for example, if the object is seen
through a refracting medium. But although the correlation is
rough, it is very useful in establishing a first approximation to the
co-ordinates of events in physical space-time. The subsequent
corrections are effected by means of causal laws, of which the
refraction of light may again serve as an example.

There is no logical reason why there should be such causal
laws, or a known relation establishing such a four-dimensional
order among events. The usual argument for the acceptance of
physical laws is that they are the simplest hypotheses hitherto
devised that are consistent with observation wherever observation
is possible. They are not, however, the only hypotheses consistent
with observation. Nor is it clear by what right we objectify our
preference for simple laws.

What physics says about the world is much more abstract than
it seems to be, because we imagine that its space is what we know
in our own experience, and that its matter is the kind of thing that
feels hard when we touch it. In fact, even assuming physics true,
what we know about the physical world is very little. Let us first
consider theoretical physics in the abstract, and then in relation
to experience.

As an abstract system, physics, at present, says something like
this: there is a manifold, called the manifold of events, which has
a system of relations among its terms by means of which it acquires
a certain four-dimensional geometry. There is an extra-geo-
metrical quantity called “energy”, which is unevenly distributed
throughout the manifold, but of which some finite amount exists
in every finite volume. The total of energy is constant. The laws
of physics are laws as to the changes in the distribution of energy.
To state these laws, we have to distinguish two kinds of regions,
those that are called “empty”, and those that are said to contain
“matter”. There are very small material systems called “atoms”;
each atom may contain any one of a certain discrete denumerable
series of amounts of energy. Sometimes it suddenly parts with a
finite amount of energy to the non-material environment, some-
times it suddenly absorbs a finite amount from the environment.
The laws as to these transitions from one energy level to another
are only statistical. In a given period of time, if not too short,
there will be, in a given state of the environment, a calculable

345

human knowledge: its scope and limits

number of transitions of each possible kind, the smaller transitions
being commoner than the greater.

In “empty space” the laws are simpler and more definite.
Parcels of energy that leave an atom spread outward equally in all
directions, travelling with the velocity of light. Whether a parcel
travels in waves or in little units or in something which is a com-
bination of both, is a matter of convention. Everything proceeds
simply until the radiant energy hits an atom, and then the atom
may absorb a finite amount of it, with the same individual inde-
terminancy and statistical regularity as applies to the emission of
energy by atoms.

The amount of energy emitted by an atom in a given transition
determines the “frequency” of the radiant energy that results.
And this in turn determines the kinds of effects that the radiant
energy can have upon any matter that it may encounter. “Fre-
quency” is a word associated with waves, but if the wave theory
of light is discarded “frequency” may be taken as a measurable
but undetermined quality of a radiation. It is measurable by its
effects.

So much for theoretical physics as an abstract logical system.
It remains to consider how it is connected with experience.

Let us begin with the geometry of space-time. We assume that
the position of a point in space-time can be determined by four
real numbers, called co-ordinates; it is also generally supposed,
though this is not essential, that to every set of four real numbers
as co-ordinates (if not too great) a position in space-time corre-
sponds. It will simplify exposition to adopt this supposition. If
we do, the number of positions in space-time is the same as the
number of real numbers, which is called c. Now of every class of c
entities we can assert every kind of geometry in which there is a
one-one correspondence between a position and a finite ordered
set of real numbers (co-ordinates). Therefore to specify the
geometry of a manifold tells us nothing unless the ordering
relation is given. Since physics is intended to give empirical
truth, the ordering relation must not be a purely logical one, such
as might be constructed in pure mathematics, but must be a
relation defined in terms derived from experience. If the ordering
relation is derived from experience, the statement that space- time
has such-and-such a geometry is one having a substantial empirical
content, but if not, not.

346

SPACE-TIME AND CAUSALITY

I suggest that the ordering relation is contiguity or compresence,
in the sense in which we know these in sensible experience. Some-
thing must be said about these.

Contiguity is a property given in sight and touch. Two portions
of the visual field are contiguous if their apparent distances and
their angular co-ordinates (up-and-down, right-and-left) differ
very little. Two parts of my body are contiguous if the qualities
by which I locate a touch in the two parts differ very little. Conti-
guity is quantitative, and therefore enables us to make series of
percepts: if A and B and C are contiguous, but B is more conti-
guous to both A and C than they are to each other, B is to be put
between A and C. There is also contiguity in time. When we
hear a sentence, the first and second words are more contiguous
than the first and third words. In this way, by means of spatial
and temporal contiguity, our experiences can be arranged in an
ordered manifold. We may assume that this ordered manifold is
a part of the ordered manifold of physical events, and is ordered
by the same relation.

For my part, however, I prefer the relation of “compresence”.
If we use this relation, we suppose that every event occupies a
finite amount of space-time, that is to say, no event is confined
to a point of space or an instant of time. Two events are said to be
“compresent” when they overlap in space-time; this is the defini-
tion for abstract physics. But we need, as we saw, a definition
derived from experience. As an ostensive definition from exper-
ience I should give the following: Two events are “compresent”
when they are related in the way in which two simultaneous parts
of one experience are related. At any given moment, I am seeing
certain things, hearing others, touching others, remembering
others, and expecting yet others. All these percepts, recollections,
and expectations are happening to me now ; I shall say that they
are mutually “compresent”. I assume that this relation, which
I know in my own experience, can also hold between events that
are not experienced, and can be the relation by which space-time
order is constructed. This will have as a consequence that two
events are compresent when they overlap in space- time, which,
if space-time order is taken as already determined, may serve,
within physics , as the definition of compresence.

Compresence is not the same thing as simultaneity, though it
implies it. Compresence, as I mean it, is to be taken as known

347

HUMAN knowledge: its scope and limits

through experience, and having only an ostensive definition. Nor
should I define “compresence” as “ simultaneity in one person's
experience”. I should object to this definition on two grounds:
first, that it could not be extended to physical occurrences ex-
perienced by no one; second, that “experience” is a vague word.
I should say that an event is “experienced” when it gives rise to
a habit, and that broadly speaking this only happens if the event
occurs where there is living matter. If this is correct, “experience”
is not a fundamental concept.

The question now arises: can we construct space-time order
out of compresence alone, or do we need something further? Let
us take a simplified hypothesis. Suppose there are n events, a l9
a 2 , . . . a n9 and suppose a x is compresent only with a 2 , a 2 is
compresent with a x and a 3 , a 3 , with a 2 and a 4 , and so on. We can
then construct the order a v a 2l . . . a n . We shall say that an event
is “between” two others if it is compresent with both, but they
are not compresent with each other; and, more generally, if
a t b 9 c are three different events, we shall say that b is “between”
a and c if the events compresent with both a and c are a proper
part of the events compresent with b . This may be taken as the
definition of “between”. Supplemented by suitable axioms, it
will generate the kind of order we want.

It should be observed that we cannot construct space-time
order out of Einstein's relation of “interval”. The interval between
two parts of a light- ray is zero, and yet we have to distinguish
between a light-ray that goes from A to B and one that goes
from B to A. This shows that “interval” alone does not
suffice.

If the above point of view is adopted, points in space-time
become classes of events. I have dealt with this subject in “The
Analysis of Matter”, and in Chapters VI and VIII of this Part,
and will therefore say no more about it.

So much for the definition of space-time order in terms of
experience. It remains to restate the connection of physical
events in the outer world with percepts.

When energy emitted by matter as a result of quantum transi-
tions travels, without further quantum transitions, to a given
part of a human body, it sets up a train of quantum transitions
which ultimately reach the brain. Assuming the maxim “same
cause, same effect”, with its consequence, “different effects,

348

SPACE-TIME AND CAUSALITY

different causes”, it follows that, if two trains of radiant energy,
falling on the same point of the body, cause different percepts,
there must be differences in the two trains, and therefore in the
quantum transitions that gave rise to them. Assuming the existence
of causal laws, this argument seems unobjectionable, and gives a
basis for the inference from perceptions to the material source
of the process by which they are caused.

I think — though I say this with hesitation — that the distinction
between spatial and temporal distance requires the consideration
of causal laws. That is to say, if there is a causal law connecting
an event A with an event B, then A and B are separated in time,
and it is a matter of convention whether we shall also consider
them separated in space. There are, however, some difficulties
about this view. A number of people may hear or see something
simultaneously, and in this case there is a causal connection with
no time- interval. But in such a case the connection is indirect,
like that connecting brothers or cousins ; that is to say, it travels
first from effect to cause and then from cause to effect. But how
are we to distinguish cause from effect before we have established
the time-order ? Eddington says we do so by means of the second
law of thermodynamics. In a spherical radiation, we take it that
it travels from a centre, not to it. But I, since I wish to connect
physics with experience, should prefer to say that we establish
time-order by means of memory and our immediate experience
of temporal succession. What is remembered is, by definition,
in the past; and there are earlier and later within the specious
present. Anything compresent with something remembered, but
not with my present experience, is also in the past. From this
starting-point we can extend the definition of time-order, and the
distinction of past and future, step by step to all events. We can
then distinguish cause from effect, and say that causes are always
earlier than effects.

According to the above theory, there are certain elements that
are carried over unchanged from the world of sense to the world
of physics. These are: the relation of compresence, the relation
of earlier and later, some elements of structure, and differences in
certain circumstances — i.e. when we experience different sensa-
tions belonging to the same sense, we may assume' that their
causes differ. This is the residue of naive realism that survives in
physics. It survives primarily because there is no positive argu-

349

HUMAN knowledge: its scope and limits

ment against it, because the resulting physics fits the known
facts, and because prejudice causes us to cling to naive realism
wherever it cannot be disproved. Whether there are any
better reasons than these for accepting physics remains to be
examined.

35 °

PART V

PROBABILITY

INTRODUCTION

I T is generally recognized that the inferences of science and
common sense differ from those of deductive logic and
mathematics in a very important respect, namely, that, when
the premisses are true and the reasoning correct, the conclusion
is only probable . We have reasons for believing that the sun will
rise to-morrow, and everybody is agreed that, in practice, we can
behave as if these reasons justified certainty. But when we examine
them we find that they leave some room, however little, for doubt.
The doubt that is justified is of three sorts. As regards the first
two : on the one hand there may be relevant facts of which we are
ignorant; on the other hand, the laws that we have to assume
in order to predict the future may be untrue. The former reason
for doubt does not much concern us in our present inquiry, but
the latter is one which demands detailed investigation. But there
is a third kind of doubt, which arises when we know a law to
the effect that something happens usually, or perhaps in an over-
whelming majority of instances, though not always; in this case
we have a right to expect what is usual, though not with
complete confidence. For example, if a man is throwing dice,
it very seldom happens that he throws double sixes ten times
running, although this is not impossible; we have therefore a
right to expect that he will not do so, but our expectation ought
to be tinged with doubt. All these kinds of doubt involve some-
thing that may be called “probability”, but this word is capable
of different meanings, which it will be important to us to dis-
entangle.

Mathematical probability arises always from a combination of
two propositions, of which one may be completely known, while
the other is completely unknown. If I draw a card from a pack,
what is the chance that it will be an ace ? I know completely the
constitution of a pack of cards, and I am aware that one card
in thirteen is an ace ; but I am completely ignorant as to which
card I shall draw. But if I say “probably Zoroaster existed”, I
am saying something about the degree of uncertainty, or of
credibility, attaching to the one proposition “Zoroaster existed”.
This is quite a different concept from that of mathematical
probability, although in many cases the two are correlated.

353

human knowledge: its scope and limits

Science is concerned to infer laws from particular facts. An
inference of this sort cannot be deductive, unless, in addition to
particular facts, there are general laws among our premisses; as
a matter of pure logic this is fairly evident. It is sometimes thought
that, though particular facts cannot make a general law certain ,
they can make it probable . Particular facts can certainly cause belief
in a general proposition; it is our experience of particular men
dying that has caused us to believe that all men are mortal. But
if we are justified in believing that all men are mortal, that must
be because, as a general principle, certain kinds of particular facts
are evidence of general laws. And since deductive logic knows
no such principle, any principle which will justify inference from
the particular to the general must be a law of nature, i.e. a state-
ment that the actual universe has a certain character which it
would be possible for it not to have. I shall undertake the search
for some such principle or principles in Part VI ; in Part V I shall
only contend that induction by simple enumeration is not such
a principle, and unless severely restricted is demonstrably invalid.

We infer, in science, not only laws, but also particular facts.
If we read in the newspaper that the King is dead, we infer that
he is dead; if we find that we shall have to make a long railway
journey without a chance of a meal, we infer that we shall be
hungry. All such inferences can only be justified if it is possible
to ascertain laws. If there were not general laws, every man’s
knowledge would be confined to what he himself has experienced.
It is more necessary that there should be laws than that they
should be known. If A is always followed by B, and an animal,
seeing A, expects B, the animal may be said to know that B is
coming without having knowledge of the general law. But although
some knowledge of facts as yet unperceived can be acquired in
this way, it is impossible to get far without knowledge of general
laws. Such laws, in general, state probabilities (in one sense), and
are themselves only probable (in another). E.g. it is probable
(in one sense) that if you have cancer it is probable (in
another sense) that you will die. This state of affairs makes
it evident that we cannot understand scientific method without
a previous investigation of the different kinds of probability.

Although such an investigation is necessary, I do not think
that probability has quite the importance attached to it by some
authors. The importance that it has arises in two ways. On the

354

INTRODUCTION

one hand, we need, among the premisses of science, not only
data derived from perception and memory, but also certain
principles of synthetic inference, which cannot be established
by deductive logic or by arguments from experience, since they
are presupposed in all inference from experienced facts to other
facts or to laws. These premisses may be admitted to be in some
degree uncertain, i.e. to have not the highest “degree of credi-
bility”. It will be part of our analysis of this form of probability
to maintain, in spite of Keynes's adverse opinion, that data and
inferential premisses may be uncertain. This is one way in which
the theory of probability is relevant, but there is also another.
It appears that we frequently know (in some sense of the word
“know") that something happens usually, but perhaps not always
— e.g. that lightning is followed by thunder. In that case, we have
a class of cases A, of which we have reason to believe that most
belong to the class B. (In our illustration, A is times shortly after
lightning, and B is times when thunder is audible.) In such
circumstances, given an instance of the class A concerning which
we do not know whether it is an instance of the class B, we have
a right to say that it is “probably” a member of the class B. Here
“probably” has not the meaning that it has when we are speaking
of degrees of credibility, but the quite different meaning that it
has in the mathematical theory of probability.

For these reasons, and also because probability- logic is much
less complete and incontrovertible than elementary logic, it is
necessary to develop the theory of probability in some detail, and
to examine various controversial questions of interpretation. It
is to be remembered that the whole discussion of probability is
of the nature of prolegomena to the investigation of the postulates
of scientific inference.

355

Chapter I

KINDS OF PROBABILITY

attempts to establish a logic of probability have been
L\ numerous, but to most of them there have been fatal
L ^objections. One of the causes of faulty theories has been
failure to distinguish — or rather, determination to confound —
essentially different concepts which, so far as usage goes, have
an equal right to be called “probability”. I propose, in this
Chapter, to make a preliminary exploration of these different
concepts in a discursive manner, leaving to later Chapters the
attempt to reach precise definitions.

The first large fact of which we have to take account is the
existence of the mathematical theory of probability. There is,
among mathematicians who have concerned themselves with this
theory, a fairly complete agreement as to everything that can be
expressed in mathematical symbols, but an entire absence of
agreement as to the interpretation of the mathematical formulae.
In such circumstances, the simplest course is to enumerate the
axioms from which the theory can be deduced, and to decide that
any concept which satisfies these axioms has an equal right, from
the mathematician’s point of view, to be called “probability”. If
there are many such concepts, and if we are determined to choose
between them, the motives of our choice must lie outside mathe-
matics.

There is one very simple concept which satisfies the axioms
of the theory of probability, and which is on other accounts
advantageous. Given a finite class B which has n members, and
given that m of these belong to some other class A, then we say
that, if a member of B is chosen at random, the chance that it
will belong to the class A is m/w. Whether this definition is
adequate to the uses that we wish to make of the mathematical
theory of probability, is a question which we shall have to inves-
tigate at a later stage; if it is not, we shall have to look for some
other interpretation of mathematical probability.

It must be understood that there is here no question of truth
or falsehood. Any concept which satisfies the axioms may be
taken to be mathematical probability. In fact, it might be desirable

356

KINDS OF PROBABILITY

to adopt one interpretation in one context, and another in another,
for convenience is the only guiding motive. This is the usual
situation in the interpretation of a mathematical theory. For
example, as we have seen, all arithmetic can be deduced from
five axioms enumerated by Peano, and therefore, if all we want
of numbers is that they should obey the rules of arithmetic, we
may define as the series of natural numbers any series satisfying
Peano’s five axioms. Now these axioms are satisfied by any pro-
gression, and, in particular, by the series of natural numbers
starting, not with o, but with ioo, or 1,000, or any other finite
integer. It is only if we decide that we want our numbers to serve
for enumeration, not only for arithmetic, that we have a motive
for choosing the series that starts with o. Similarly, in the case
of the mathematical theory of probability, the interpretation to
be chosen may depend upon the purpose we have in view.

The word “probability” is often used in ways that are not, or
at least not obviously, capable of interpretation as the ratio of
the numbers of two finite classes. We may say : Probably Zoroaster
existed, probably Einstein’s theory of gravitation is better than
Newton’s, probably all men are mortal . 1 In these cases, we might
maintain that there is evidence of a certain kind, which is known
to be conjoined with a conclusion of a certain kind in a large
majority of cases; in this way, theoretically, the definition of
probability as the ratio of the numbers of two classes might
become applicable. It is possible, therefore, that instances such
as the above do not involve a new meaning of “probability”.

There are, however, two dicta which we are all inclined to
accept without much examination, but which, if accepted, involve
an interpretation of “probability” which it seems impossible to
reconcile with the above definition. The first of these dicta is
Bishop Butler’s maxim that “probability is the guide of life”. The
second is the maxim that all our knowledge is only probable,
which has been specially emphasized by Reichenbach.

Bishop Butler’s maxim is obviously valid according to one very
common interpretation of “probability”. When — as is usually the
case — I am not certain what is going to happen, but I must act
upon one or other hypothesis, I am generally well advised to
choose the most probable hypothesis, and I am always well
advised to take account of probability in making my decision.

1 Not to be confused with “all men are probably mortal”.

357

HUMAN knowledge: its scope and limits

But there is an important logical difference between this kind
of probability and the mathematical kind, namely that the latter
is concerned with propositional functions 1 and the former with
propositions. When I say that the chance of a coin coming heads
is a half, that is a relation between the two propositional functions
“ x is a toss of the coin” and “ x is a toss of the coin which comes
heads”. If I am to infer that, in a particular case, the chance
of heads is a half, I must state that I am considering the particular
case solely as an instance. If I could consider it in all its par-
ticularity, I should, in theory, be able to decide whether it will
fall heads or fall tails, and I should no longer be in the domain
of probability. When we use probability as a guide to conduct,
it is because our knowledge is inadequate; we know that the
event in question is one of a class B of events, and we may know
what proportion of this class belongs to some class A in which
we are interested. But the proportion will vary according to our
choice of the class B ; we shall thus obtain different probabilities,
all equally valid from the mathematical standpoint. If probability
is to be a guide in practice, we must have some way of selecting
one probability as the probability. If we cannot do this, all the
different probabilities remain equally valid, and we shall be left
without guidance.

Let us take an instance, in which every sensible man is guided
by probability, I mean life insurance. I ascertain the terms on
which some company is willing to insure my life, and I have to
decide whether insurance on these terms is likely to prove an
advantageous bargain, not to insurers in general, but to me.

My problem is different from that of the insurance company,
and much more difficult. The insurance company is not interested
in my individual case: it offers insurance to all members of a
certain class, and need only take account of statistical averages.
But I may believe that I have special reasons to expect a long
life, or that I am like the Scotchman who died the day after
completing his insurance, remarking with his last breath “I always
was a lucky fellow”. Every circumstance of my health and my
way of life is relevant, but some of these may be so uncommon
that I can get no reliable help from statistics. At last I decide to

1 I.e. with sentences containing an undefined variable — e.g. “A is a
man” — which become propositions when we assign a value to the
variable — in the above case “A”.

358

KINDS OF PROBABILITY

consult a medical man, who, after a few questions, remarks
genially: “Oh, I expect you’ll live to be 90”. I am painfully aware,
not only that his judgment is slap-dash and unscientific, but also
that he wishes to please me. The probability at which I finally
arrive is thus something quite vague and quite incapable of
numerical measurement; but it is upon this vague probability
that, as a disciple of Bishop Butler, I have to act.

The probability which is the guide of life is not the mathe-
matical kind, not only because it is not relative to arbitrary data,
but to all data that bear on the question at issue, but also because
it has to take account of something which lies wholly outside the
province of mathematical probability, and which may be called
“intrinsic doubtfulness”. This is what is relevant when it is said
that all our knowledge is only probable. Consider for example
a distant memory which has grown so dim that we can no longer
trust it with any confidence, a star so faint that we are not sure
whether we really see it, or a noise so slight that we think it may
be only imagined. These are extreme cases, but in a lesser degree
the same sort of doubtfulness is very common. If we assert, as
Reichenbach does, that all our knowledge is doubtful, we cannot
define this doubtfulness in the mathematical way, for in the com-
piling of statistics it is assumed that we know whether or not
this A is a B, e.g. whether this insured person has died. Statistics
are built up on a structure of assumed certainty as to past
instances, and a doubtfulness which is universal cannot be merely
statistical.

I think, therefore, that everything we feel inclined to believe
has a “degree of doubtfulness”, or, inversely, a “degree of
credibility”. Sometimes this is connected with mathematical
probability, sometimes not; it is a wider and vaguer conception.
It is not, however, purely subjective. There is a cognate subjective
conception, namely, the degree of conviction that a man feels
about any of his beliefs, but “credibility”, as I mean it, is objective
in the sense that it is the degree of credence that a rational man
will give. When I add up my accounts, I give some credence to
the result the first time, considerably more if I get the same result
the second time, and almost full conviction if I get it a third time.
This increase of conviction goes with an increase of evidence,
and is therefore rational. In relation to any proposition about
which there is evidence, however inadequate, there is a cor-

359

human knowledge: its scope and limits

responding “degree of credibility”, which is the same as the
degree of credence given by a man who is rational. (This latter
may perhaps be regarded as a definition of the word “rational”.)
The importance of probability in practice is due to its connec-
tion with credibility, but if we imagine this connection to be
closer than it is, we bring confusion into the theory of
probability.

The connection between credibility and subjective conviction
is one that can be studied empirically; we need not, therefore,
have any views on this subject in advance of the evidence. A
conjurer, for instance, can arrange circumstances in a manner
known to himself but calculated to deceive his audience ; he can
thus acquire data as to how to cause untrue convictions, which
are likely to be useful in advertising and propaganda. We cannot
so easily study the relation of credibility to truth, because we
commonly accept a high degree of credibility as sufficient evidence
of truth, and if we do not do so we can no longer discover any
truths. But we can discover whether propositions having high
credibility form a mutually consistent set, since the set contains
the propositions of logic.

I suggest, as a result of the above preliminary discussion, that
two different concepts each, on the basis of usage, have an equal
claim to be called “probability”. The first of these is mathematical
probability, which is numerically measurable and satisfies the
axioms of the probability- calculus ; this is the sort that is involved
in the use of statistics, whether in physics, in biology, or in the
social sciences, and is also the sort that we hope is involved in
induction. This sort of probability has to do always with classes,
not with single cases except when they can be considered merely
as instances.

But there is another sort, which I call “degree of credibility”.
This sort applies to single propositions, and takes account always
of all relevant evidence. It applies even in certain cases in which
there is no known evidence. The highest degree of credibility
to which we can attain applies to most perceptive judgments;
varying degrees apply to judgments of memory, according to their
vividness and recentness. In some cases the degree of credibility
can be inferred from mathematical probability, in others it
cannot; but even when it can it is important to remember that
it is a different concept. It is this sort, and not mathematical

360

KINDS OF PROBABILITY

probability, that is relevant when it is said that all our know-
ledge is only probable, and that probability is the guide of
life.

Both kinds of probability demand discussion. I shall begin with
mathematical probability.

361

Chapter II

MATHEMATICAL PROBABILITY

I N this Chapter I propose to treat the theory of probability
as a branch of pure mathematics, in which we deduce the
consequences of certain axioms without seeking to assign this
or that interpretation 1 to them. It is to be observed that, while
interpretation, in this field, is controversial, the mathematical
calculus itself commands the same measure of agreement as any
other branch of mathematics. This situation is in no way peculiar.
The interpretation of the infinitesimal calculus was for nearly
two hundred years a matter as to which mathematicians and
philosophers debated; Leibniz held that it involved actual infini-
tesimals, and it was not till Weierstrass that this view was definitely
disproved. To take an even more fundamental example: there has
never been any dispute as to elementary arithmetic, and yet the
definition of the natural numbers is still a matter of controversy.
We need not be surprised, therefore, that there is doubt as to
the definition of probability though there is none (or very little)
as to the calculus of probability.

Following Johnson and Keynes, we will denote by “ pjh ” the
undefined notion: “The probability of p given A”. When I say
that this notion is undefined, I mean that it is only defined by
the axioms or postulates about to be enumerated. Anything satis-
fying these axioms is an “interpretation” of the calculus of
probability, and it is to be expected that there will be many
possible interpretations. No one of these is more correct or more
legitimate than another, but some may be more important than
others. So, in finding an interpretation of Peano’s five axioms for
arithmetic, the interpretation in which the first number is o is
more important than that in which it is 3781 ; it is more important
because it enables us to identify the interpretation of the formalist
conception with the conception recognized in enumeration. But
for the present we will ignore all questions of interpretation, and
proceed with the purely formal treatment of probability.

The axioms or postulates required are given in much the same

1 As to “interpretation”, see Part IV, Chapter I.
362

MATHEMATICAL PROBABILITY

way by different authors. The following statement is taken from
Professor C. D. Broad. 1 The axioms are:

I. Given p and A, there is only one value of />/A. We can
therefore speak of “ the probability of p given A”.

II. The possible values of p/h are all the real numbers from
o to i, both included. (In some interpretations we confine the
possible values to rational numbers; this is a question I shall
discuss later.)

III. If A implies p, then />/A = i. (We use “i” to denote
certainty.)

IV. If A implies not-/), then pjh = o. (We use “o” to denote
impossibility.)

V. The probability of both p and q given A is the probability
of p given A multiplied by the probability of q given p and A,
and is also the probability of q given A multiplied by the probability
of p given q and A.

This is called the “conjunctive” axiom.

VI. The probability of p and/or q given A is the probability
of p given A plus the probability of q given A minus the probability
of both p and q given A.

This is called the “disjunctive” axiom.

It is immaterial, for our purposes, whether these axioms are
all necessary ; what concerns us is only that they are sufficient.

Some observations are called for as regards these axioms. It
is obvious that II, III, and IV embody, in part, conventions
which might easily be changed. If, when they are adopted, the
measure of a given probability is x , we might equally well adopt
as its measure any number / (#) which increases as x increases ;
for i and o in III and IV we should then substitute /(i) and / (o).

According to the above axioms, a proposition which must be
true if the data are true is to have the probability i in relation
to the data, and one which must be false if the data are true is
to have the probability o in relation to the data.

It is important to observe that our fundamental concept p/h
is a relation of two propositions (or conjunctions of propositions),
not a property of a single proposition p. This distinguishes
probability as it occurs in the mathematical calculus from proba-
bility as required as a guide to practice, for the latter kind has
to belong to a proposition in its own right, or, at least, in relation
1 Mind, N.S. No. 210, p. 98.

363

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

to data which are not arbitrary, but determined by the problem
and the nature of our knowledge. In the calculus, on the contrary,
the choice of the data A is wholly arbitrary.

Axiom V is the “conjunctive” axiom. It gives the chance that
each of two events will happen. For example: if I draw two cards
from a pack, what is the chance that they are both red? Here
“A” represents the datum that the pack consists of 26 red cards
and 26 black ones; “p” stands for the statement “the first card
is red” and “5” for the statement “the second card is red”. Then
“(p and #)/A” is the chance that both are red, p/h is the chance
that the first is red, “<7/ ( p and A)” is the chance that the second is
red, given that the first is red. Obviously p /A = 1 /2, q/(p and A) =
25/51. Thus by the axiom the chance that both are red is \ • ff .

Axiom VI is the “disjunctive” axiom. In the above illustration,
it gives the chance that at least one of the cards is red. It says
that the chance that at least one is red is the chance that the first
is red plus the chance that the second is red (when it is not given
whether the first is red or not) minus the chance that both are red.
This is £ + i — £ * 5 1 » using the result obtained above by the
use of the conjunctive axiom.

It is obvious that, by means of Axioms V and VI, given the
separate probabilities of any finite collection of events, we can
calculate the probability of their all happening, or of at least one
of them happening.

From the conjunctive axiom it follows that

pl(q and A) =

(p/h) X ql(p and h)

q/h

This is called the “principle of inverse probability”. Its utility
may be illustrated as follows. Let p be some general theory, and
q an experimental datum relevant to p. Then p/h is the probability
of the theory p on the previously known data, qjh is the probability
of q on the previously known data, and qj{p and A) is the proba-
bility of q if p is true. Thus the probability of the theory p after q
has been ascertained is got by multiplying the previous probability
of p by the probability of q given p, and dividing by the previous
probability of q . In the most useful case, the theory p will be
one which implies 5, so that qj{p and A) = 1. In that case,

PKS and h) — ^

364

MATHEMATICAL PROBABILITY

That is to say, the new datum q increases the probability of p
in proportion to the antecedent improbability of q. In other words,
if our theory implies something very astonishing, and the astonish-
ing thing is then found to happen, that greatly increases the
probability of our theory.

This principle may be illustrated by the discovery of Neptune
regarded as a confirmation of the law of gravitation. Here
p = the law of gravitation, h = all relevant facts known before
the discovery of Neptune, q = the fact that Neptune was found
in a certain position. Thus qjh was the antecedent probability
that a hitherto unknown planet would be found in a certain small
region of the heavens. Let us take this to be m/w. Then after the
discovery of Neptune the probability of the law of gravitation
was njm times as great as before.

It is obvious that this principle is of great importance in judging
the bearing of new evidence on the probability of a scientific
theory. We shall find, however, that it proves somewhat dis-
appointing, and does not yield such good results as might have
been hoped.

There is an important proposition, sometimes called Bayes’s
theorem, which is as follows. Let p lt p 2 > . . . . p n be n mutually
exclusive possibilities, of which some one is known to be true;
let h be the general data, and q some relevant fact. We wish to
know the probability of one possibility />„ given q y when we know
the probability of each p r before q was known, and also the
probability of q given />„ for every r. We have

p,l(q and h) = [?/(/>, and h) . />,/*] /2[?/(/», and h) . pjh]

This proposition enables us, for example, to solve the following
problem : We are given n + i bags, of which the first contains
n black balls and no white ones, the second contains n — i black
balls and one white one, the (r + i)** contains n — r black balls
and r white ones. One bag is chosen, but we do not know which;
m balls are drawn from it, and are found to be all white; what is
the probability that the 1 th bag has been chosen? Historically,
this problem is important in connection with Laplace’s pretended
proof of induction.

Take next Bernoulli’s law of large numbers. This states that
if, on each of a number of occasions, the chance of a certain event

365

HUMAN knowledge: its scope and limits

occurring is p , then, given any two numbers 8 and c, however
small, the chance that, from a certain number of occasions
onward, the proportion of occasions on which the event occurs
will ever differ from p by more than e, is less than 8.

Let us illustrate by the case of tossing a coin. We suppose that
heads and tails are equally probable. I say that, in all likelihood,
after you have tossed often enough, the proportion of heads will
never again depart from 1/2 by more than c, however small e
may be; I say further that, however small 8 may be, the chance
of such a departure anywhere after the n th toss is less than 8,
provided n is sufficiently great.

As this proposition is of great importance in the applications
of probability, for instance to statistics, let us spend a little longer
in familiarizing ourselves with exactly what it asserts in the above
case of tossing a coin. I assert first, let us say, that from some
point onwards, the percentage of heads will always remain between
49 and 51. You dispute my assertion, and we decide to test it
empirically as far as this is possible. The theorem then asserts
that, the longer we go on, the more likely we are to find my
assertion borne out by the facts, and that, as the number of tosses
is increased, this likelihood approaches certainty as a limit. You
are convinced by experiment, we will suppose, that from some
point onwards the percentage of heads remains always between
49 and 51, but I now assert that, from some further point
onwards, it will always remain between 49*9 and 50*1. We
repeat our experiment, and again you are convinced after a
time, though probably after a longer time than before. After any
given number of throws, there is a chance that my assertion may
not be verified, but this chance diminishes as the number of
throws increases, and can be made less than any assigned chance,
however small, by going on long enough.

The above proposition is easy to deduce from the axioms, but
cannot, of course, be adequately tested empirically, since it
involves infinite series. If the tests we can make seem to confirm
it, the opponent can always say they would not have done so if
we had gone on longer; and if they seem not to confirm it, the
supporter of the theorem can equally say that we have not gone
on long enough. The theorem cannot, therefore, be either proved
or disproved by empirical evidence.

The above are the principal propositions in the pure theory

366

MATHEMATICAL PROBABILITY

of probability that are important in our inquiry. I will, however,
say something more on the subject of the n + i bags, containing
n balls each, some white and some black, the i ,A bag con-
taining r white balls and n — r black balls. The data are as follows :
I know that the bags have these varying numbers of white and
black balls, but there is no way of distinguishing them from the
outside. I choose one bag at haphazard, and draw from it, one
by one, m balls which I do not replace after drawing them. They
turn out to be all white. In view of this fact, I want to know two
things: first, what is the chance that I have chosen the bag that
has only white balls? second, what is the chance that the next
ball I draw will be white ?

We proceed as follows. Let h be the fact that the bags are
constituted in the above manner, and q the fact that m white
balls have been drawn ; also let p r be the hypothesis that we have
chosen the bag containing r white balls. It is obvious that r must
be at least as great as m, i.e.

If r is less than m , then pjqh = o and qjpji = o.

After some calculation, it turns out that the chance that we have

chosen the bag in which all the balls are white is —

n + i

We now want to know the chance that the next ball will be
white. After some further calculation, it turns out that this

. m + 1
chance is — — .

m + 2

Note that this is independent of «, and that, if m is large, it is
very nearly i .

I have not, in the above outline, included any arguments on
the subject of induction, which I postpone to a later stage. I shall
first consider the adequacy of a certain interpretation of proba-
bility, in so far as this can be considered independently of the
problems connected with induction.

367

Chapter III

THE FINITE-FREQUENCY THEORY

I N this Chapter we are concerned with a certain very simple
interpretation of “probability”. We have, first, to show that
it satisfies the axioms of Chapter II, and then to consider,
in a preliminary way, how far it can be made to cover ordinary
uses of the word “probability”. I shall call it “the finite frequency
theory”, to distinguish it from another form of frequency theory
which we shall consider later.

The finite frequency theory starts from the following definition:
Let B be any finite class, and A any other class. We want to
define the chance that a member of B chosen at random will be
a member of A, e.g. that the first person you meet in the street
will be called Smith. We define this probability as the number
of B’s that are A’s divided by the total number of B’s. We denote
this by the symbol A/B.

It is obvious that a probability so defined must be a rational
fraction or o or i .

A few illustrations will make the purport of this definition clear.
What is the chance that an integer less than io, chosen at random,
will be a prime? There are 9 integers less than 10, and 5 of them
are primes; therefore the chance is 5/9. What is the chance that
it rained in Cambridge on my birthday last year, assuming that
you do not know when my birthday is ? If m was the number of
days on which it rained, the chance is mj 365. What is the chance
that a man whose name occurs in the London telephone book
will be called Smith ? To solve this problem, you must first count
the entries under the name “Smith”, and then count all the entries,
and divide the former number by the latter. What is the chance
that a card drawn at random from a pack will be a spade?
Obviously 13/52, i.e. 1/4. If you have drawn a spade, what is
the chance that you will draw another? The answer is 12/51.
In a throw of two dice, what is the chance that the numbers will
add up to 8 ? There are 36 ways in which the dice may fall, and
in 5 of these the numbers add up to 8, so the chance is 5/36.

It is obvious that, in a number of elementary cases, the above
definition gives results that accord with usage. Let us now inquire
whether probability, so defined, satisfies the axioms.

368

THE FINITE-FREQUENCY THEORY

The letters p and q and A, which occur in the axioms, must
now be taken to stand for classes or propositional functions, not
for propositions. Instead of “ A implies />” we shall have “A is
contained in />”; “ p and q” will stand for the common part of
the two classes p and q , while “ p or q” will be the class of all terms
that belong to either or both of the two classes p and q .

Our axioms were :

I. There is only one value of pjh. This will be true unless A is
null, in which case pjh = o/o. We shall therefore assume that A
is not null.

II. The possible values of pjh are all the real numbers from
o to i. In our interpretation, they will be only the rational num-
bers, unless we can find a way of extending our definition to
infinite classes. This cannot be done simply, since division does
not yield a unique result when the numbers concerned are infinite.

III. If A is contained in p , then pjh = i. In this case, the
common part of A and p is A, therefore the above follows from
our definition.

IV. If A is contained in not-/), then pjh = o. This is obvious
on our definition, since in this case the common part of A and p
is null.

V. The conjunctive axiom . — This states, on our interpretation,
that the proportion of members of A which are members of both
p and q is the proportion of members of A that are members of
p multiplied by the proportion of members of p and A that are
members of q. Suppose the number of members of A is a, the
number of members common to p and A is A, and the number
of members common to p and q and A is c . Then the proportion
of members of A that are members of both p and q is cja\ the
proportion of members of A that are members of p is bja t and
the proportion of members of p and A that are members of q is
cjb. Thus our axiom is verified, since c/a — bja X cjb .

VI. The disjunctive axiom . This says, on our present inter-

pretation, keeping the above meanings of a , A, and c , and adding
that d is the number of members of A that are members of p or q
or both, while e is the number of members of A that are members
of j, that ,

d b * e C • J L i

- = - 4 , i.e. d= A + e — c

a a a a

which again is obvious.

369

HUMAN knowledge: its scope and limits

Thus our axioms are satisfied if h is a finite class which is not
null, except that the possible values of a probability are confined
to rational fractions.

It follows that the mathematical theory of probability is valid
on the above interpretation.

We have, however, to inquire as to the scope of probability so
defined, which is, prima facie , much too narrow for the uses that
we wish to make of probability.

In the first place, we wish to be able to speak of the chance
that some definite event will have some characteristic, not only
of the chance that an unspecified member of a class will have it.
For example: You have already made a throw with two dice, but
I have not seen the result. What is, for me, the probability that
you have thrown double sixes? We want to be able to say that
it is 1/36, and if our definition does not allow us to say so, it is
inadequate. In such a case, we should say that we are considering
an event merely as an instance of a certain class; we should say
that, if a is considered merely as a member of the class B, the
chance that it belongs to the class A is A/B. But it is not very
clear what is meant by “considering a definite event merely as
a member of a certain class”. What is involved in such a case
is this: We are given some characteristic of an event which, to
more complete knowledge than ours, is sufficient to determine it
uniquely; but relatively to our knowledge, we have no way of
finding out whether it belongs to the class A, though we do know
that it belongs to the class B. You, who have thrown the dice,
know whether the throw belongs to the class of double sixes, but
I do not know this. My only relevant knowledge is that it is one
of the 36 possible kinds of throws. Or take the following question:
What is the chance that the tallest man in the United States lives
in Iowa? Somebody may know who he is; at any rate there is a
known method of finding out who he is. If this method has been
successfully employed, there is a definite answer not involving
probability, namely, either that he does live in Iowa or that he
does not. But I have not this knowledge. I can ascertain that the
population of Iowa is m and that of the United States is n , and
say that, relative to these data, the probability that he lives in
Iowa is m/n. Thus when we speak of the probability of a definite
event having some characteristic, we must always specify the data
relative to which the probability is to be estimated.

370

THE FINITE-FREQUENCY THEORY

We may say generally: Given any object a , and given that a
is a member of the class B, we say that, in relation to this datum,
the probability that a is an A is A/B as previously defined. This
conception is useful because we often know enough about some
object to enable us to define it uniquely, without knowing enough
to determine whether it has this or that property. “The tallest
man in the United States” is a definite description, which applies
to one and only one man, but I do not know what man, and
therefore for me it is an open question whether he lives in Iowa.
“The card I am about to draw” is a definite description, and in
a moment I shall know whether this description applies to a red
or a black card, but as yet I do not know. It is this very common
condition of partial ignorance as to definite objects that makes
it useful to apply probability to definite objects, and not only to
wholly undefined members of classes.

Although partial ignorance is what makes the above form of
probability useful , ignorance is not involved in the concept of
probability, which would still have the same meaning for omnis-
cience as for us. Omniscience would know whether a is an A,
but would still be able to say: Relative to the datum that a is
a B, the probability that a is an A is A/B.

In the application of our definition to a definite instance, there
is a possible ambiguity in certain cases. To make this clear, we
must use the language of properties rather than classes. Let the
class A be defined by the property <p and the class B by the
property ip. Then we say :

The probability that a has the property cp given that it has the
property ip is defined as the proportion of things having both the
properties (p and \p to those having the property tp. We denote
“ a has the property <p” by “(pa”. But if a occurs more than once
in “<pa” y there will be an ambiguity. E.g. suppose “<pa” is “a
commits suicide”, i.e. “a kills a”. This is a value of “x kills x” y
which is the class of suicides; also of “a kills x” y which is the
class of persons whom a kills ; also of “x kills a” y which is the
class of persons who kill a . Thus in defining the probability of
<pa y if “a” occurs more than once in “<pa” y we must indicate
which of its occurrences are to be regarded as values of a variable
and which not.

It will be found that we can interpret all elementary theorems
in accordance with the above definition.

371

HUMAN knowledge: its scope and limits

Take, for example, Laplace’s supposed justification of induction :

There are N + i bags, each containing N balls.

Of the bags, the r + i th contains r white balls and N — r black
balls. We have drawn from one bag n balls, all white. What is
the chance

(a) that we have chosen the bag in which all are white?

( b ) that the next ball will be white ?

Laplace says that (a) is ( n + i) /(N + i) and ( b ) is ( n + i )/(» + 2).

Let us illustrate by some numerical instances. First: Suppose
there are 8 balls altogether, of which 4 have been drawn, all
white. What are the chances (a) that we have chosen the bag
consisting only of white balls, and ( b ) that the next ball drawn
will be white?

Let p 9 represent the hypothesis that we have chosen the bag
with r white balls. The data exclude p 0 , p x> /> 2 , /> 3 . If we have /> 4 ,
there is only one way in which we can have drawn 4 whites, and
there remain 4 ways of drawing a black, none of drawing a white.
If we have p 5l there were 5 ways in which we could have drawn
4 whites, and for each of these there was 1 way of drawing
another white, and 3 of drawing a black; thus from p & we get
a contribution of 5 cases where the next is white and 15 where
it is black. If we have /> 6 , there were 15 ways of choosing 4 whites,
and when they had been chosen there remained 2 ways of choosing
a white and 2 of choosing a black ; thus we get from p Q 30 cases
of another white and 30 where the next is black. If we have p ly
there are 35 ways of drawing 4 whites, and after they have been
drawn there remain 3 ways of drawing a white and one of drawing
a black; thus we get 105 ways of drawing another white and 35
of drawing a black. If we have /> 8 , there are 70 ways of drawing
4 whites, and when they have been drawn there are 4 ways of
drawing another white and none of drawing a black ; thus we get
from p s 280 cases of a fifth white and none of a black. Adding,
we have 5 + 30 + 105 + 280, i.e. 420, cases in which the fifth
ball is white, and 4 -f 15 + 30 + 35, i.e. 84, cases in which the
fifth ball is black. Therefore the odds in favour of white are
420 to 84, i.e. s to 1 ; that is to say, the chance of the fifth ball
being white is 5/6.

The chance that we have chosen the bag in which all the balls
are white is the ratio of the number of ways of choosing 4 white
balls from this bag to the total number of ways of choosing 4

37 *

THE FINITE-FREQUENCY THEORY

white balls. The former, we have seen, is 70; the latter is
1 + S + IS + 3S + 70, i.e. 126. Therefore the chance is 70/126,
i.e. 5/9.

Both these results are in accordance with Laplace’s formula.
To take one more numerical example: suppose there are 10
balls, of which 5 have been drawn, and have been found to be
all white. What is the chance of p 10 , i.e. of our having chosen
the bag with only white balls? And what is the chance that the
next ball will be white ?

p 5 possible in 1 way; if p 5> no way of another white, 5 of a black.

p 6 >>

„ 6ways;„/> 6 , 1

>»

> 4

Pi »»

„ 21 „ ; 2 ways

»»

, 3

Ps »»

»» 56 >> > uPhi 3 »>

>»

, 2

P 9 »»

,, 126 ,, ; ,,/> 9 , 4 „

»»

, 1

P 10

»» 252 ,, , , , P 10 > 5 »»

»>

, 0

Thus the chance of p 10 is 252/[i + 6 + 21 + 56 + 126 + 252]
252/462, i.e. 6/1 1.

The ways in which the next ball can be white are

6 + 21 X 2 + 56 X 3 + 126 X 4 + 252 X 5, i.e. 1980
and the ways in which it can be black are

5 + 4x6 + 3x21 + 2x56+ 126, i.e. 330.

i.e.

Therefore the odds in favour of white are 1,980 to 330, i.e. 6 to 1,
so that the chance of another white is 6/7. This again is in
accordance with Laplace’s formula.

Let us now take Bernoulli’s law of large numbers. We may
illustrate it as follows : suppose we toss a coin n times, and put 1
for every time it comes heads, 2 for every time it comes tails,
thus forming a number of n digits. We will suppose every possible
sequence to come just once. Thus if n = 2, we have the four
numbers 11, 12, 21, 22; if n = 3, we have the 8 numbers in,
112, 121, 122, 211, 212, 221, 222; if n = 4, we have 16 numbers,
mi, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121,
2122, 2211, 2212, 2221, 2222;
and so on. Taking the last of the above lists, we find
1 number all 1 ’s

4 numbers with three 1 ’s and one 2
6 „ „ two Ts „ two 2’s

4 „ „ one 1 „ three 2’s

1 number all 2*s.

373

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

These numbers, i, 4, 6, 4, 1, are the coefficients in (a+ 6) 4 .
It is easy to prove that, for n digits, the corresponding numbers
are the coefficients in (a + b) n . All that Bernoulli’s theorem
amounts to is that, if n is large, the sum of the coefficients near
the middle is very nearly equal to the sum of all the coefficients
(which is z n ). Thus if we take all possible series of heads and
tails in a large number of tosses, the immense majority have very
nearly the same number of both ; the majority and the nearness,
moreover, increase indefinitely as the number of throws increases.

Though Bernoulli’s theorem is more general and more precise
than the above statements with equi-probable alternatives, it is
to be interpreted, on our present definition of “probability”, in
a manner analogous to the above. It is a fact that, if we form all
numbers that consist of 100 digits, each of which is either 1 or 2,
about a quarter have 49 or 50 or 51 digits that are 1, nearly a half
have 48 or 49 or 50 or 51 or 52 digits that are 1, more than half
have from 47 to 53 digits that are 1, and about three-quarters
from 46 to 54. As the number of digits is increased, the pre-
ponderance of cases in which i’s and 2’s are nearly evenly
balanced increases.

Why this purely logical fact should be regarded as giving us
good ground for expecting that, when we toss a penny a great
many times, we shall in fact attain an approximately equal number
of heads and tails, is a different question, involving laws of nature
in addition to logical laws. I mention it now only to emphasize
the fact that I am not at present discussing it.

I want to lay stress on the fact that, in the above interpretation,
there is nothing about possibility, and nothing which essentially
involves ignorance. There is merely a counting of members of
a class B and determining what proportion of them also belong
to a class A.

It is sometimes contended that we need an axiom of equi-
probability — e.g. to the effect that heads and tails are equally
probable. If this means that in fact they occur with approximately
equal frequency, the assumption is not necessary to the mathe-
matical theory, which, as such, is not concerned with actual
occurrences.

Let us now consider possible applications of the finite-frequency
definition to cases of probability which might seem to fall out-
side it.

374

THE FINITE-FREQUENCY THEORY

First : in what circumstances can the definition be extended to
infinite collections? Since we have defined a probability as a
fraction, and since fractions are meaningless when numerator and
denominator are infinite, it will only be possible to extend the
definition when there is some means of proceeding to a limit.
This requires that the a" s, of which we are to estimate the proba-
bility of their being V s, should form a series, in fact a progression,
so that they are given as a lt Og, a 3 , . . . a nf . . . where for every
finite integer n there is a corresponding a n and vice versa. We can
then denote by the proportion of a ' s up to a n that belong to b.
If, as n increases, p n approaches a limit, we can define this limit
as the probability that an a will be a b } We must, however,
distinguish the case in which the value of p n oscillates about the
limit from that in which it approaches the limit from one side
only. If we repeatedly toss a coin, the number of heads will be
sometimes more than half the total, sometimes less; thus p n
oscillates about the limit 1/2. But if we consider the proportion
of primes up to w, this approaches the limit zero from one side
only: for any finite n , p n is a definite positive fraction, which, for
large values of «, is approximately 1 /log n. Now 1 /log n approaches
zero as n increases indefinitely. Thus the proportion of primes
approaches zero, but we cannot say “no integers are primes ,, ;
we may say that the chance of an integer being a prime is infini-
tesimal, but not zero. Obviously the chance of an integer being
a prime is greater than that of its being (say) both odd and even,
although the chance is less than any finite fraction, however small.
I should say that, when the chance that an a is a b is strictly zero,
we can infer “no a is a b”, but when the chance is infinitesimal
we cannot make this inference.

It is to be observed that, unless we make some assumption
about the course of nature, we cannot use the method of pro-
ceeding to the limit when we are dealing with a series which is
defined empirically. For example, if we toss a given coin repeatedly,
and find that the number of heads, as we go on, approaches
continually nearer to the limit 1/2, that does not entitle us to
assume that this really would be the limit if we could make our
series infinite. It may be, for example, that, if n is the number

1 This limit depends upon the order of the a* s, and therefore belongs
'to them as a series, not as a class.

375

HUMAN knowledge: its scope and limits

of tosses, the proportion of heads does not approximate strictly
to 1/2 but to

where N is a number much larger than any that we can reach
in actual experiments. In that case, our inductions would begin
to be empirically falsified just as we were thinking they were
firmly established. Or again, it might happen, with any empirical
series, that after a time it became utterly lawless, and ceased in
any sense to approach a limit. If, then, the above extension to
infinite series is to be used in empirical series, we shall have to
invoke some kind of inductive axiom. Without this, there is no
reason for expecting the later parts of such a series to continue
to exemplify some law which the earlier parts obey.

In ordinary empirical judgments of probability, such, for
example, as are contained in the weather forecast, there is a
mixture of different elements which it is important to separate.
The simplest hypothesis — unduly simplified for purposes of illus-
tration — is that some symptom is observed which, in (say) 90
per cent of the cases in which it has been previously observed,
has been followed by rain. In that case, if inductive arguments
were as indubitable as deductive ones, we should say “there is
a 90 per cent probability of rain”. That is to say, the present
moment belongs to a certain class (that of moments when the
symptom in question is present) of which 90 per cent are moments
preceding rain. This is probability in the mathematical sense
which we have been considering. But it is not this alone that
makes us uncertain whether it will rain. We are also uncertain
as to the validity of the inference; we do not feel sure that the
symptom in question will, in the future, be followed by rain nine
times out of ten. And this doubt may be of two kinds, one scientific,
the other philosophical. We may, while retaining full confidence
in scientific procedure in general, feel that, in this case, the data
are too few to warrant an induction, or that not sufficient
care has been taken to eliminate other circumstances which may
have also been present and may be more invariable precursors
of rain. Or, again, the records may be doubtful: they may have
been rendered nearly indecipherable by rain, or have been made
by a man who was shortly afterwards certified as insane. Such
doubts are within scientific procedure, but there are also the

376

THE FINITE-FREQUENCY THEORY

doubts raised by Hume: is inductive procedure valid, or is it
merely a habit which makes us comfortable ? Any or all of those
reasons may make us hesitant about the 90 per cent chance of
rain which our evidence inclines us to believe in.

We have, in cases of this sort, a hierarchy of probabilities. The
primary level is: Probably it will rain. The secondary level is:
Probably the symptoms I noticed are a sign of probable rain. The
tertiary level is: Probably certain kinds of events make certain
future events probable. Of these three levels, the first is that of
common sense, the second that of science, and the third that of
philosophy.

In the first stage, we have observed that, hitherto, A has been
followed, nine times out of ten, by B; in the past, therefore
A has made B probable in the sense of finite frequency. We
suppose without reflection, at this stage, that we may expect the
same thing in the future.

In the second stage, without questioning the general possibility
of inferring the future from the past, we realize that such in-
ferences should be submitted to certain safeguards, such, for
example, as those of Mill’s four methods. We realize also that
inductions, even when conducted according to the best rules, are
not always verified. But I think our procedure can still be brought
within the scope of the finite frequency theory. We have made
in the past a number of inductions, some more careful, some
less so. Of those made by a certain procedure, a proportion p
have, so far, been verified ; therefore this procedure, hitherto, has
conferred a probability p upon the inductions that it sanctioned.
Scientific method consists largely of rules by means of which p
(as tested by the past results of past inductions) can be made
to approach nearer to 1 . All this is still within the finite frequency
theory, but it is now inductions that are the single terms in our
estimate of frequency.

That is to say, we have two classes A and B, of which A consists
of inductions that have been performed in accordance with certain
rules, and B consists of inductions which experience hitherto has
confirmed. If n is the number of members of A, and m is the
number of members common to A and B, then m/n is the chance
that an induction conducted according to the above rules will
have, up to the present, led to results which, when they could
be tested, were found to be true.

377

HUMAN knowledge: its scope and limits

In saying this, we are not using induction; we are merely
describing a feature of the course of nature so far as it has been
observed. We have, however, found a criterion of the excellence
(hitherto) of any suggested rules of scientific procedure, and we
have found it within the finite frequency theory. The only novelty
is that our units are now inductions, not single events. The
inductions are treated as occurrences, and it is only those that
have actually occurred that are to be regarded as members of
our class A.

But as soon as we argue either that an individual induction
which has hitherto been confirmed will, or will probably, be con-
firmed in the future, or that rules of procedure, which have given,
so far, a large proportion of inductions that have been confirmed
so far, are likely to give a large proportion of confirmed inductions
in the future, we have passed outside the finite frequency theory,
since we are dealing with classes of which the numbers are not
known. The mathematical theory of probability, like all pure
mathematics, though it gives knowledge, does not (at least in one
important sense) give anything new ; induction, on the other hand,
certainly gives something new, and the only doubt is whether
what it gives is knowledge.

I do not want, as yet, to examine induction critically; I wish
only to make clear that it cannot be brought within the scope of
the finite frequency theory, even by the device of considering
a particular induction as one of a class of inductions, since tested
inductions can only supply inductive evidence in favour of a
hitherto untested induction. If, then, we say that the principle
which validates induction is “probable”, we must be using the
word “probable” in a different sense from that of the finite
frequency theory; the sense in question must, I should say, be
what we called “degree of credibility”.

I incline to think that, if induction, or whatever postulate we
may decide upon as a substitute, is assumed, all precise and
measurable probabilities can be interpreted as finite frequencies.
Suppose I say, for example: “There is a high probability that
Zoroaster existed”. To substantiate this statement, I shall have
to consider, first, what is the alleged evidence in his case, and
then to look out for similar evidence which is known to be either
veridical or misleading. The class upon which the probability
depends is not the class of prophets, existent and non-existent,

378

THE FINITE-FREQUENCY THEORY

for by including the non-existent we make the class somewhat
vague ; nor can it be the class of existent prophets only, since the
question at issue is whether Zoroaster belongs to this class. We
shall have to proceed as follows : There is, in the case of Zoroaster,
evidence belonging to a certain class A; of all the evidences that
belong to this class and can be tested, we find that a proportion p
are veridical; we therefore infer, by induction, that there is a
probability p in favour of the similar evidence in the case of
Zoroaster. Thus frequency plus induction covers this use of
probability.

Or suppose we say, like Bishop Butler: “It is probable that
the universe is the result of design on the part of a Creator”.
Here we start with such subsidiary arguments as that a watch
implies a watchmaker. There are very many instances of watches
known to be made by watchmakers, and none of watches known
to be not made by watchmakers. There is in China a kind of
marble which sometimes, by accident, produces what appear to
be pictures made by artists; I have seen the most astonishing
examples. But this is so rare that, when we see a picture, we are
justified (assuming induction) in inferring an artist with a very
high degree of probability. What remains for the episcopal
logician, as he emphasizes by the title of his book, is to prove
the analogy; this may be held doubtful, but cannot well be
brought under the head of mathematical probability.

So far, therefore, it would seem that doubtfulness and mathe-
matical probability — the latter in the sense of finite frequency —
are the only concepts required in addition to laws of nature and
rules of logic. This conclusion, however, is only provisional.
Nothing definitive can be said until we have examined certain
other suggested definitions of “probability”.

379

Chapter IV

THE MISES-REICHENBACH THEORY

T he frequency interpretation of probability, in a form
different from that of the previous chapter, has been set
forth in two important books, both by German professors
who were then in Constantinople. 1

Reichenbach’s work is a development of that of v. Mises, and
is in various ways a better statement of the same kind of theory.
I shall therefore confine myself to Reichenbach.

After giving the axioms of the probability calculus, Reichenbach
proceeds to offer an interpretation which seems to be suggested
by the case of statistical correlations. He supposes two series
(#i» x & • • • x n » • • •)> (^d ^2* • • • y n y • • •)» an d two classes O and P.
Some or all of the x's belong to the class O; what interests him
is the question : how often do the corresponding y’s belong to
the class P ?

Suppose, for example, you were investigating the question
whether a man is predisposed to suicide by having a nagging wife.
In this case, the x’s are wives, the ys are husbands, the class O
consists of naggers, and the class P of suicides. Then given that
a wife belongs to the class O, our question is: how often does
her husband belong to the class P ?

Consider the sections of the two series consisting of the first n
terms of each. Suppose that, among the first n x’s, there are a
terms belonging to the class O, and suppose that, of these, there
are b terms such that the corresponding y belongs to the class P.
(The corresponding y is the one with the same suffix.) Then we
say that, throughout the section from x x to x ny the “relative
frequency” of O and P is bja . (If all the tf’s belong to the class O,
a = n, and the relative frequency is b/n.) We denote this relative
frequency by “H n ( 0 , P)”.

We now proceed to define “the probability of P given O”
which we denote by “W( 0 , P)”. The definition is:

1 Richard von Mises, Wahrscheinlichkeit , Statistik und Wahrheit y
2nd ed. Vienna, 1936 (1st ed. 1928); Hans Reichenbach, Wahrschein -
lichkeitslehre , Leiden, 1935. See also the latter’s Experience and Prediction ,
1938.

380

THE MISES-REICHENBACH THEORY

W(0, P) is the limit of H n (0, P) as n is indefinitely in-
creased.

This definition can be considerably simplified by the use of
a little mathematical logic. In the first place, it is unnecessary to
have two series. For both are assumed to be progressions, and there
is therefore some one-one correlator of their terms. If this
is S, to say that a certain y belongs to a class P is equivalent
to saying that the corresponding x belongs to the class of terms
having the relation S to some one or other of the members
of P. E.g. let S be the relation of wife to husband; then if y
is a married man and x is his wife, to say that y is a govern-
ment official is true if, and only if, x is the wife of a government
official.

In the second place, there is no advantage in admitting
the case in which not all the x’s belong to the class O. The
definition is only appropriate if an infinite number of the x's
belong to the class O; in that case, those that belong to O form
a progression, and the rest can be forgotten. Thus we retain
what is essential in Reichenbach ’s definition if we substitute the
following:

Let Q be a progression, and a some class, of which, in the
important cases, there are members, in the series of Q, later than
any given member. Let m be the number of members of a among
the first n members of Q. Then W(Q, a) is defined as the limit
of m/n when n is indefinitely increased.

Perhaps through inadvertence, Reichenbach speaks as if the
concept of probability were only applicable to progressions, and
had no application to finite classes. I cannot think he intends this.
The human race, for example, is a finite class, and we wish to
apply probability to vital statistics, which would be impossible
according to the letter of the definition. As a matter of psycho-
logical fact, when Reichenbach speaks of the limit for n = infinity,
he is thinking of the limit as some number which is very nearly
approached whenever n is large from an empirical point of view,
i.e. when it is not far short of the maximum that our means of
observation enable us to reach. He has an axiom or postulate to
the effect that, when there is such a number for every large
observable n , it is approximately equal to the limit for n = infinity.
This is an awkward axiom, not only because it is arbitrary, but
because most of the series with which we are concerned outside

381

HUMAN knowledge: its scope and limits

pure mathematics are not infinite; indeed it may be doubted
whether any of them are. We are in the habit of assuming that
space-time is continuous, which implies the existence of infinite
series; but this assumption has no basis except mathematical
convenience.

I shall assume, in order to make Reichenbach *s theory as
adequate as possible, that, where finite classes are concerned,
the definition of the last chapter is to be retained, and that the
new definition is only intended as an extension enabling us to
apply probability to infinite classes. Thus his H w (O, P) will be
a probability, but one applying only to the first n terms of the
series.

What Reichenbach postulates, as his form of induction, is
something like this: Suppose we have made N observations as
to the correlation of O and P, so that we are in a position to
calculate H n (O, P) for all values of n up to n = N, and suppose
that, throughout the last half of the values of n , H w (O, P) always
differs from a certain fraction p by less than c, where e is small.
Then it shall be posited that, however much we were to increase
w, H w (O, P) would still lie within these narrow boundaries, and
therefore W (O, P), which is the limit for n = infinity, will also
lie within these boundaries. Without this assumption, we can
have no empirical evidence as to the limit for n = infinity, and
the probabilities for which the definition is specially designed
must remain totally unknown.

In defence of Reichenbach’s theory, in face of the above
difficulties, two things may be said. In the first place, he may
contend that it is not necessary to suppose n to approach infinity
indefinitely; for all practical purposes, it suffices if n is allowed
to become very large. Suppose, for instance, that we are dealing
with vital statistics. It does not matter to an insurance company
what will happen to the statistics if they are prolonged for another
10,000 years; at most, the next ioo years concern it. If, when
we have accumulated statistics, we assume that frequencies will
remain roughly the same until we have ten times as many data
as we have now, that is enough for almost all practical purposes.
Reichenbach may say that, when he speaks of infinity, he is using
a convenient mathematical shorthand, meaning only “a good deal
more of the series than we have investigated hitherto”. The case
is exactly analogous, he might say, to that of the empirical deter-

382

THE MISES-REICHENBACH THEORY

mination of a velocity. In theory, a velocity can only be deter-
mined if there is no limit to the smallness of measureable spaces
and times; in practice, since there is such a limit, the velocity
at an instant can never be known even approximately. We can,
it is true, know with a fair measure of accuracy the average
velocity throughout a short time. But even if we assume a postulate
of continuity, the average velocity throughout (say) a second gives
absolutely no indication as to the velocity at a given instant during
that second. All motion might consist of periods of rest separated
by instants of infinite velocity. Short of this extreme hypothesis,
and even if we assume continuity in the mathematical sense, no
finite velocity at an instant is incompatible with any finite average
velocity throughout a finite time, however short, which contains
that instant. For practical purposes, however, this is of no con-
sequence. Except in a few phenomena such as explosions, if we
take the velocity at any instant through a very short measurable
time to be approximately the average velocity during that time,
the laws of physics are found to be verified. “Velocity at an in-
stant”, therefore, may be regarded as nothing but a convenient
mathematical fiction.

In like manner, Reichenbach may say, when he speaks of the
limit of a frequency when n is infinite, he means only the actual
frequency for very large numbers, or rather this frequency with
a small margin of error. The infinite and the infinitesimal are
equally unobservable, and therefore (he may say) equally irrelevant
to empirical science.

I am inclined to admit the validity of this answer. I only regret
that I do not find it explicitly in Reichenbach ’s books; I think,
nevertheless, that he must have had it in mind.

The second point in favour of his theory is that it applies to
just the sort of cases in which we wish to use probability argu-
ments. We wish to use these arguments when we have some data
as to a certain future event, but not enough to determine its
character in some respect that interests us. My death, for example,
is a future event, and if I am insuring my life I may wish to
know what evidence exists as to the likelihood of death occurring
in some given year. In such a case we always have a number of
individual facts recorded in a series, and we assume that the
frequencies we have found hitherto will more or less continue.
Or take gambling, from which the whole subject took its rise.

HUMAN knowledge: its scope and limits

We are not interested in the mere fact that there are 36 possible
results of a throw with two dice. What we are interested in is the
fact (if it be a fact) that in a long series of throws each of these
36 possibilities will be realized an approximately equal number
of times. This is a fact which does not follow from the mere
existence of 36 possibilities. When you meet a stranger, there are
exactly two possibilities: on the one hand, he may be called
Ebenezer Wilkes Smith ; on the other hand, he may not. But in
a long life, during which I have met a great many strangers, I
have only once found the former possibility realized. The pure
mathematical theory, which merely enumerates possible cases, is
devoid of practical interest unless we know that each possible
case occurs approximately with equal frequency, or with some
known frequency. And this, if we are considering events, not a
logical schema, can only be known through actual statistics, the
use of which, it may be said, must proceed more or less as in
Reichenbach’s theory.

This argument, also, I shall admit provisionally; it will be
examined afresh when we come to consider induction.

There is an objection of a quite different kind to Reichenbach’s
theory as he states it, and that has to do with his introducing
series where only classes seem to be logically relevant. Let us
take an illustration : what is the chance that an integer chosen at
random will be a prime ? If we take the integers in their natural
order, the chance, on his definition, is zero; for if n is an integer,
the number of primes less than or equal to n is approximately
nflogn if n is large, so that the chance of an integer less than n
being a prime approximates to 1 /logw, and the limit of 1 /log n as
i» is indefinitely increased is zero. But now suppose we rearrange
the integers on the following plan: Put first the first 9 primes,
then the first number that is not a prime, then the next 9 primes,
and then the second number that is not a prime, and so on
indefinitely. When the integers are arranged in this order,
Reichenbach’s definition shows that the chance of a number
selected at random being a prime is We could even arrange
the integers so that the chance of a number not being a prime
would be zero. To get this result, begin with the first non-prime,
i.e. 4, and put after the n th number which is not a prime the n
primes next after those already placed; this series begins 4, 1,6,

*, 3 * 8 > Sf 7» 9* I 3> l 7 > l 9 > 33 > *<>> 29, 31, 37, 41, 43, 12

THE MISES-REICHENBACH THEORY

In this arrangement, there will be, before the (n + i) tt non-
prime, n non-primes and J » (» + i) primes; thus as n increases,
the ratio of the number of non-primes to the number of primes
approaches o as a limit.

From this illustration it is obvious that, if Reichenbach’s
definition is accepted, given any class A having as many terms
as there are natural numbers, and given any infinite sub-class B,
the chance that an A selected at random will be a B will be
anything from o to i (both included), according to the way in
which we choose to distribute the B’s among the A’s.

It follows that, if probability is to apply to infinite collections,
it must apply to series, not to classes. This seems strange.

It is true that, where empirical data are concerned, they are all
given in a time-order, and therefore as a series. If we choose to
assume that there is going to be an infinite number of events of
the kind we are investigating, then we can also decide that our
definition of probability is to apply only so long as the events
are arranged in temporal sequence. But outside pure mathematics
no series are known to be infinite, and most are, as far as we can
judge, finite. What is the chance that a man of 60 will die of
cancer? Surely we can estimate this without assuming that the
number of men who, before time ends, will have died of cancer,
is infinite. But according to the letter of Reichenbach’s definition
this should be impossible.

If probabilities depend upon taking events in their temporal
order, rather than in any other order of which they are susceptible,
then probability cannot be a branch of logic, but must be part
of the study of the course of nature. This is not Reichenbach’s
view; he holds, on the contrary, that all true logic is probability-
logic, and that the classical logic is at fault because it classifies
propositions as true or false, not as having this or that degree
of probability. He should, therefore, be able to state what is
fundamental in probability-theory in abstract logical terms, with-
out introducing accidental features of the actual world, such as
time.

There is great difficulty in combining a statistical view of
probability with the view, which Reichenbach also holds, that
all propositions are only probable in varying degrees that fall
short of certainty. The difficulty is that we seem committed to
an endless regress. Suppose we say it is probable that a man

385

HUMAN knowledge: its scope and limits

This argument, also, I shall admit provisionally; it will be
examined afresh when we come to consider induction.

There is an objection of a quite different kind to Reichenbach’s
theory as he states it, and that has to do with his introducing
series where only classes seem to be logically relevant. Let us
take an illustration : what is the chance that an integer chosen at
random will be a prime ? If we take the integers in their natural
order, the chance, on his definition, is zero ; for if n is an integer,
the number of primes less than or equal to n is approximately
n/logn if n is large, so that the chance of an integer less than n
being a prime approximates to i/logw, and the limit of i/log n as
n is indefinitely increased is zero. But now suppose we rearrange
the integers on the following plan: Put first the first 9 primes,
then the first number that is not a prime, then the next 9 primes,
and then the second number that is not a prime, and so on
indefinitely. When the integers are arranged in this order,
Reichenbach’s definition shows that the chance of a number
selected at random being a prime is We could even arrange
the integers so that the chance of a number not being a prime
would be zero. To get this result, begin with the first non-prime,
i.e. 4, and put after the n th number which is not a prime the n
primes next after those already placed; this series begins 4, 1,6,

*, 3» 8 » 5. 7. *«. 9, 13. 17. 19. 23. i°» 29, 31. 37. 4*. 43. **

384

THE MISES-REICHENBACII THEORY

In this arrangement, there will be, before the (n -f i) tt non-
prime, n non-primes and J n (n + i) primes; thus as n increases,
the ratio of the number of non- primes to the number of primes
approaches o as a limit.

It follows that, if probability is to apply to infinite collections,
it must apply to series, not to classes. This seems strange.

385 BB

human knowledge: its scope and limits

who has plague will die of it. This means that, if we could
ascertain the whole series of men who, from the earliest times
till the extinction of the human race, will have suffered from
plague, we should find that more than half of them will have died
of it. Since the future and much of the past are unrecorded, we
assume that the recorded cases are a fair sample. But now we
are to remember that all our knowledge is only probable ; therefore
if, in compiling our statistics, we find it recorded that Mr. A
had plague and died of it, we must not regard this item as certain,
but only as probable. To find out how probable it is, we must
include it in a series, say of official death certificates, and we must
find some means of ascertaining what proportion of death cer-
tificates are correct. Here a single item in our statistics will be:
“Mr. Brown was officially certified to have died, but turned out
to be still alive”. But this, in turn, is to be only probable, and
must therefore be one of a series of recorded official errors, some
of which turned out to be not errors. That is to say, we must
collect cases where it was falsely believed that a person certified
dead had been found to be still alive. To this process there can
be no end, if all our knowledge is only probable, and probability
is only statistical. If we are to avoid an endless regress, and if
all our knowledge is to be only probable, “probability” will have
to be interpreted as “degree of credibility”, and will have to
be estimated otherwise than by statistics. Statistical probability
can only be estimated on a basis of certainty, actual or
postulated.

I shall return to Reichenbach in connection with induction.
For the present, I wish to make clear my own view as to the
connection of mathematical probability with the course of nature.
Let us take as an illustration a case of Bernoulli's law of large
numbers, choosing the simplest possible case. We have seen that,
if we make up all possible integers consisting of n digits, each
either i or 2, then, if n is large — say not less than 1,000 — a vast
majority of the possible integers have an approximately equal
number of i’s and 2’s. This is merely an application of the fact
that, in the binomial expansion of (x + y) n , when n is large the
sum of the coefficients near the middle falls not far short of the
sum of all the coefficients, which is 2 n . But what has this to do
with the statement that, if I toss a penny often, I shall probably
get an approximately equal number of heads and tails ? The one

386

THE MISES-REICHENBACH THEORY

is a logical fact, the other, apparently, an empirical fact; what is
the connection between them ?

With some interpretations of “probability”, a statement con-
taining the word “probable” can never be an empirical statement.
It is admitted that what is improbable may happen, and what
is probable may fail to happen. It follows that what does happen
does not show that a previous judgment of probability was either
right or wrong; every imaginable course of events is logically
compatible with every imaginable anterior estimate of probabili-
ties. This can only be denied by maintaining that what is very
improbable does not happen, which we have no right to maintain.
In particular, if induction asserts only probabilities, then what-
ever may happen is logically compatible both with the truth and
with the falsehood of induction. Therefore the inductive principle
has no empirical content. This is a reductio ad absurdum , and
shows that we must connect the probable with the actual more
closely than is sometimes done.

If we adhere to the finite frequency theory — and so far I have
seen no reason for not doing so — we shall say that, if we assert
“a is an A” to be probably given “a is a B”, we mean that, in
fact, most members of B are members of A. This is a statement
of fact, but not a statement about a. And if I say that an inductive
argument (suitably formulated and limited) makes its conclusion
probable, I mean that it is one of a class of arguments, most of
which have conclusions that are true.

What, now, can I mean when I say that the chance of heads
is a half? To begin with, this, if true, is an empirical fact; it does
not follow from the fact that, in tossing a coin, there are only
two possibilities, heads and tails. If it did, we could infer that
the chance of a stranger being called Ebenezer Wilkes Smith is
a half, since there are only two alternatives, that he is so called
or that he isn’t. With some coins, heads come oftener than tails;
with others, tails oftener than heads. When I say, without speci-
fying the coin, that the chance of heads is a half, what do I mean ?

My assertion, like all other empirical assertions that pretend
to numerical exactitude, must be only approximate. When I say
that a man’s height is 6 ft. i in., I am allowed a margin of
error; even if I have said it on oath, I cannot be convicted of
perjury if it turns out that I am a hundredth of an inch out.
Similarly, I must not be held to have made a false statement

387

HUMAN knowledge: its scope and limits

about the penny if it turns out that 0*500001 would have been
a more accurate estimate than 0*5. It is doubtful, however,
whether any evidence could make me think 0*500001 a better
estimate than 0*5. In probability, as elsewhere, we take the
simplest hypothesis which approximately fits the facts. Take (say)
the law of falling bodies. Galileo made a certain number of obser-
vations, which fitted more or less with the formula s = \gt 2 . No
doubt he could have found a function f(t) such that s = f(t) would
have fitted his observations more exactly, but he preferred a
simple formula which fitted well enough. 1 In the same way, if
I tossed a coin 2,000 times and got 999 heads and 1,001 tails,
I should take the chance of heads to be a half. But what exactly
should I mean by this statement ?

This question shows the strength of Reichenbach’s definition.
According to him, I mean that, if I continue long enough, the
proportion of heads will come, in time, to be permanently very
near \ ; in fact, it will come to differ from £ by less than any
fraction however small. This is a prophecy; if it is correct, my
estimate of the probability is correct, but if not, not. What can
the finite frequency theory oppose to this ?

We must distinguish between what the probability is and what
it probably is. As to what the probability is, that depends upon
the class of tosses we are considering. If we are considering tosses
with a given coin, then if, in the whole of its existence, this coin
is going to have given m heads out of a total of n tosses, the
probability of heads with that coin is m\n. If we are considering
coins in general, n will have to be the total number of tosses of
coins throughout the past and future history of the world, and
m the number of these that will have been heads. We may, to
make the problem less vast, confine ourselves to tosses this year
in England, or to tosses tabulated by students of probability. In
all these cases m and n are finite numbers, and min is the proba-
bility of heads with the given conditions.

But none of the above probabilities are known. We are there-
fore driven to make estimates of them, that is to say, to find some
way of deciding what they probably are. If we are to adhere to
the finite frequency theory, this will mean that our series of heads
and tails must be one of some finite class of series, and that we
must have some relevant knowledge about this whole class. We
1 Cf. Jeffreys, Theory of Probability , and Scientific Inference.

388

THE MISES-REICHENBACH THEORY

will suppose it to have been observed that, in every series of
10,000 or more tosses with a given coin, the proportion of heads
after the 5,ooo /A toss has never varied by more than 2€, where €
is small. We can then say : In every observed case, the proportion
of heads after 5,000 tosses with a given coin has always remained
between p — e and p + e, where p is a constant depending on
the coin. To argue from this to a case not yet observed is a matter
of induction. If this is to be valid, we shall need an axiom to the
effect that (in certain circumstances) a characteristic which is
present in all observed cases is present in a large proportion of
all cases; or, at any rate, we shall need some axiom from which
this results. We shall then be able to infer a probable probability
from observed frequencies, interpreting probability in accordance
with the finite frequency theory.

The above is only an outline suggestion of a theory. The main
point that I wish to emphasize is that, on the theory I advocate,
every probability statement (as opposed to a merely doubtful
statement) is a statement of fact , as to some proportion in a series.
In particular, the inductive principle, whether true or false, will
have to assert that, as a fact, most series of certain kinds have,
throughout, any characteristic of a certain sort which is present
in a large number of successive terms of the series. If this is a
fact, inductive arguments may yield probabilities; if not, not.
I do not at present inquire how we are to know whether it is
a fact or not; that is a problem which I shall not consider until
the last section of our inquiry.

It will be seen that, in the above discussion, we have been led
to agree with Reichenbach on many points, while consistently
disagreeing as to the definition of probability. The main objection
which I feel to his definition is that the frequency on which it
depends is hypothetical and for ever unascertainable. I disagree
also in distinguishing more sharply than he does between proba-
bility and doubtfulness, and in holding that probability- logic is
not logically the fundamental kind, as opposed to certainty-logic.

3 g 9

Chapter V

KEYNES'S THEORY OF PROBABILITY

K eynes’s Treatise on Probability (1921) sets out a theory
which is, in a sense, the antithesis of the frequency theory.
He holds that the relation used in deduction, namely
“p implies q } \ is the extreme form of a relation which might be
called “p more or less implies q”. “If a knowledge of h ”, he says,
“justifies a rational belief in a of degree a, we say there is a
probability relation of degree a between a and A”. We write
this: “ ajh = a”. “Between two sets of propositions there exists
a relation, in virtue of which, if we know the first, we can attach
to the latter some degree of rational belief.” Probability is essen-
tially a relation: “It is as useless to say 'b is probable’ as ‘A is
equal’ or ‘A is greater than’.” From “a” and “ a implies A”, we
can conclude “A”, that is to say, we can drop all mention of the
premiss and simply assert the conclusion. But if a is so related
to b that a knowledge of a renders a probable belief in b rational,
we cannot conclude anything whatever about b which has not
reference to a\ there is nothing corresponding to the dropping
of a true premiss in demonstrative inference.

Probability, according to Keynes, is a logical relation, which
cannot be defined, unless, perhaps, in terms of degrees of rational
belief. But on the whole it would seem that Keynes inclines rather
to defining “degrees of rational belief” in terms of the probability-
relation. Rational belief, he says, is derivative from knowledge:
when we have a degree of rational belief in p, it is because we
know some proposition h and also know p/A = a. It follows that
some propositions of the form “p/A = a” must be among our
premisses. Our knowledge is partly direct, partly by argument;
our knowledge by argument proceeds through direct knowledge
of propositions of the form “p implies q” or “ qjp = a”. In every
argument, when fully analysed, we must have direct knowledge
of the relation of the premisses to the conclusion, whether it be
that of implication or that of probability in some degree. Know-
ledge of A and of p/A = a leads to a “rational belief of the appro-
priate degree” in p. Keynes explicitly assumes that all direct
knowledge is certain, and that a rational belief which falls short

390

KEYNES’S THEORY OF PROBABILITY

of certainty can only arise through perception of a probability-
relation.

Probabilities in general, according to Keynes, are not numeri-
cally measurable; those that are so form a very special class of
probabilities. He holds that one probability may not be com-
parable with another, i.e. may be neither greater nor less than
the other, nor yet equal to it. He even holds that it is sometimes
impossible to compare the probabilities of p and not-/> on given
evidencj. He does not mean that we do not know enough to do
this; he means that there actually is no relation of equality or
inequality. He thinks of probabilities according to the following
geometrical scheme: Take two points, representing the o of
impossibility and the i of certainty; then numerically measurable
possibilities may be pictured as lying on the straight line between

0 and i , while others lie on various curved routes from o to i . Of
two probabilities on the same route, we can say that the one nearer

1 is the greater, but we cannot compare probabilities on different
routes, except when two routes intersect, which may happen.

Keynes needs, as we have seen, some direct knowledge of
probability-propositions. In order to make a beginning in obtain-
ing such knowledge, he examines and emends what is called the
“principle of non-sufficient reason”, or, as he prefers to call it,
the “principle of indifference”.

In its crude form, the principle states that if there is no known
reason for one rather than another of several alternatives, then
these alternatives are all equally probable. In this form, as he
points out, the principle leads to contradictions. Suppose, for
instance, you know nothing of the colour of a certain book ; then
the chances of its being blue or not blue are equal, and therefore
each is 1/2. Similarly the chance of its being black is 1/2. There-
fore the chance of its being blue or black is 1. It follows that all
books are either blue or black, which is absurd. Or suppose we
know that a certain man inhabits either Great Britain or Ireland,
shall we take these as our alternatives, or shall we take England,
Scotland, and Ireland, or shall we take each county as equally
probable? Or, if we know that the specific gravity of a certain
substance lies between 1 and 3, shall we take the intervals 1 to 2
and 2 to 3 as equally probable ? But if we consider specific volume,
the intervals 1 to 2/3 and 2/3 to 1/3 would be the natural choice,
which would make the specific gravity have equal chances of

39i

HUMAN knowledge: its scope and limits

being between i and 3/2 or between 3/2 and 3. Such paradoxes
can be multiplied indefinitely.

Keynes does not, on this account, totally abandon the principle
of indifference ; he thinks it can be so stated as to avoid the above
difficulties and still be useful. For this purpose, he first defines
“irrelevance”.

Roughly speaking, an added premiss is “irrelevant” if it does
not change the probability; i.e. A x is irrelevant in relation to
x and A if xjhfi = x/h. Thus, for example, the fact that a man’s
surname begins with M is irrelevant in estimating his chances
of death. The above definition is, however, somewhat too simple,
because h x might consist of two parts, of which one increased
the probability of x while the other diminished it. For example:
a white man’s chances of life are diminished by living in the
tropics, but are increased (or so they say) by being a teetotaller.
It may be that the death-rate among white teetotallers in the
tropics is the same as that of white men in general, but we should
not say that being a teetotaller who lives in the tropics was
irrelevant. Therefore we say that h x is irrelevant to x/h if there
is no part of h x which alters the probability of x.

Keynes now 7 states the principle of indifference in the following
form: The probabilities of a and b relative to given evidence are
equal if there is no relevant evidence relating to a without cor-
responding evidence relating to b ; that is to say, the probabilities
of a and b relative to the evidence are equal, if the evidence
is symmetrical with respect to a and b.

There is, however, still a somewhat difficult proviso to be
added. “We must exclude those cases, in which one of the alter-
natives involved is itself a disjunction of sub-alternatives of the
same form” When this condition is fulfilled, the alternatives are
called indivisible relatively to the evidence. Keynes gives a formal
definition of “divisible” as follows : An alternative <f){a) is divisible,
relatively to evidence A, if, given A, is equivalent to il <f>(b)

or $c)”, where <f>(b) and <f>(c) are incompatible, but each possi-
ble, when A is true. It is essential, here, that <£(a), <£(A), <f>(c) are all
values of the same propositional function.

Keynes thus finally accepts as an axiom the principle that, on
given evidence, <f>(a) and <f>(b) are equally probable, if (1) the
evidence is symmetrical with respect to a and A, (2) relatively to
the evidence, <f>(a) and <f>(b) are indivisible.

392

KEYNES'S THEORY OF PROBABILITY

To the above theory empiricists might raise a general objection.
They might say that the direct knowledge of probability relations
which it demands is obviously impossible. Deductive demonstrative
logic — so this argument might run — is possible because it consists
of tautologies, because it merely re- states our initial stock of
propositions in other words. When it does more than this — when,
for instance, it infers “Socrates is mortal” from “all men are
mortal”, it depends upon experience for the meaning of the word
“Socrates”. Nothing but tautologies can be known independently
of experience, and Keynes does not contend that his probability-
relations are tautologous. How, then, can they be known? For
clearly they are not known by experience, in the sense in which
judgments of perception are so known; and it is admitted that
some of them are not inferred. They would constitute, therefore,
if admitted, a kind of knowledge which empiricism holds to be
impossible.

I have much sympathy with this objection, but I do not think
we can consider it decisive. We shall find, when we come to
discuss the principles of scientific inference, that science is impos-
sible unless we have some knowledge which we could not have
if empiricism, in a strict form, were true. In any case, we should
not assume dogmatically that empiricism is true, though we are
justified in trying to find solutions of our problems which are
compatible with it. The above objection, therefore, though it may
cause a certain reluctance to accept Keynes's theory, should not
make us reject it outright.

There is a difficulty on a question which Keynes seems not
to have adequately considered, namely: Does probability in rela-
tion to premisses ever confer rational credibility on the proposition
which is rendered probable, and, if so, under what circumstances?
Keynes says that it is as nonsensical to say “ p is probable” as
to say “p is equal” or “p is greater than”. There is, according
to him, nothing analogous to the dropping of a true premiss in
deductive inference. Nevertheless, he says that, if we know A, and
we also know p/A = a, we are entitled to give to p “rational
belief in the appropriate degree”. But when we do so we are no
longer expressing a relation of p to A; we are using this relation
to infer something about p. This something we may call “rational
credibility”, and we can say; “p is rationally credible to the
degree a”. But if this is to be a true statement about p, not

393

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

involving mention of A, then A cannot be arbitrary. For suppose
pjh = a and p/h' = a , are we, supposing A and A' both known,
to give to p the degree a or the degree a of rational credibility?
It is impossible that both answers should be correct in any given
state of our knowledge.

If it is true that “probability is the guide of life”, then there
must be, in any given state of our knowledge, one probability
which attaches to p more vitally than any other, and this proba-
bility cannot be relative to arbitrary premisses. We must say that
it is the probability which results when h is taken to be all our
relevant knowledge. We can say: Given any body of propositions
constituting some person’s certain knowledge, and calling the
conjunction of this body of propositions A, there are a number
of propositions, not members of this body, which have proba-
bility-relations to it. If p is such a proposition, and pfh— a,
then a is the degree of rational credibility belonging to p for that
person. We must not say that, if h' is some true proposition,
short of A, which the person in question knows, and if p/h f = a',
then, for that person , p has the degree of credibility a'; it will
only have this degree of credibility for a person whose relevant
knowledge is summed up by A'. All this, however, no doubt
Keynes would admit. The objection is, in fact, only to a certain
looseness of statement, not to anything essential to the theory.

A more vital objection is as to our means of knowing such
propositions as pjh — a. I am not now arguing a priori that we
cannot know them; I am merely inquiring how we can. It will
be observed that if “probability” cannot be defined, there must
be probability- propositions which cannot be proved, and which,
therefore, if we are to accept them, must be among the premisses
of our knowledge. This is a general feature of all logically articu-
lated systems. Every such system starts, of necessity, with an
initial apparatus of undefined terms and unproved propositions.
It is obvious that an undefined term cannot appear in an inferred
proposition unless it has occurred in at least one of the unproved
propositions ; but a defined term need not occur in any unproved
proposition. For example: so long as there were held to be
undefined terms in arithmetic, there had to be also unproved
axioms: Peano had three undefined terms and five axioms. But
when numbers and addition are defined logically, arithmetic needs
no unproved propositions beyond those of logic. So, in our case,

394

Keynes’s theory of probability

if “probability” can be defined, it may be that all propositions
in which the word occurs can be inferred; but if it cannot be
defined, there must, if we are to know anything about it, be
propositions, containing the word, which we know without
extraneous evidence.

It is not quite clear what sort of propositions Keynes would
admit as premisses in our knowledge of probability. Do we
directly know propositions of the form “p/A — a”? And when
a probability is not numerically measurable, what sort of thing
is a? Or do we only know equalities and inequalities, i.e.
pjh <qjh t or p/A = qjh ? I incline to think that the latter is
Keynes’s view. If so, the fundamental facts in the subject are
relations of three propositions, not of two : we ought to start from
a triadic relation

P (p, q, h)

meaning: given A, p is less probable than q. We might then say:

“p/A = q/h” means : “Neither P(p, q , h) nor P(q f p, A)”.

We should assume that P is asymmetrical and transitive with
respect to p and q while h is kept constant. Keynes’s principle
of indifference, if accepted, will then enable us, in certain cir-
cumstances, to prove pjh = qjh. And from this basis the calculus
of probabilities, in so far as Keynes considers it valid, can be
built up.

The above definition of equality can only be adopted if p\h
and qjh are comparable; if (as Keynes holds possible) neither is
greater than the other and yet they are not equal, the definition
must be abandoned. We could meet this difficulty by axioms as
to the circumstances under which two probabilities must be
comparable. When they are comparable they lie on one route
between o and i. On the right-hand side of the above definition
of “p/A = qjh” we must then add that pjh and qlh are “com-
parable”.

Let us now re-state Keynes’s principle of indifference. He is
concerned to establish circumstances in which pjh = qjh. This
will happen, he says, if two conditions (sufficient but not neces-
sary) are fulfilled. Let p be of the form <f>(a) and q of the
form <f>(b) ; then h must be symmetrical with respect to a and b,
and <f>(a), <f>{b) must be “indivisible”.

395

HUMAN knowledge: its scope and limits

When we say that h is symmetrical with respect to a and 6,
we mean, presumably, that, if h is of the form f(a , b ), then

/(*, b) — a).

This will happen, in particular, if f(a , b) is of the form g(a). g(b) y
which is the case when the information that h gives about a and b
consists of separate propositions, one about a and the other about 6,
and both are values of one propositional function.

We now put p = <f>(a) } q = <f>(b ), h — f(a, b).

Our axiom must be to the effect that, with a suitable proviso,
the interchange of 0(a) and </>(b) cannot make any difference. This
involves that b) = *(*)//(«, b)

provided <f>(a) and <f>(b ) are comparable with respect to /(a , b).
This follows if, as a general principle,

0a/0 a = 0£/0 b

that is to say, if probability depends not on the particular subject
but on propositional functions. There seems hope, along these
lines, of arriving at a form of the principle of indifference which
might have more self-evidence than Keynes’s.

Let us, for this purpose, examine his condition of indivisibility.
Keynes defines “0(a) is divisible” as meaning that there are
two arguments b and c such that “0a” is equivalent to “ <f>b or
and <f>b and (j>c cannot both be true, while <f>b y (f>c are both
possible given h. I do not think this is quite what he really wishes
to say. We get nearer to what he wishes, I think, if we assume
a and b and c to be classes, of which a is the sum of b and c.
In that case, </> must be a function which takes classes as argu-
ments. E.g. let a be an area on a target, divided into two parts
b and c. Let “ <f>a ” be “some point of a is hit” and “0a” be
“some point of a is aimed at”. Then (f>a is divisible in the above
sense, and we do not have

fa/tpa — <j>b jifjb

for obviously <j>a jifja is greater than <f>b/ifjb.

But it is not clear that our earlier condition, namely that h
should be symmetrical with respect to a and b y does not suffice.
For now h contains the proposition “6 is part of a”, which is
not symmetrical.

396

KEYNES’S THEORY OF PROBABILITY

Keynes discusses the conditions for fa/ipa = <f>b/tfjb t and gives
as an example of failure the case where <]>x . = . x is Socrates.
In that case, no matter what ipx may be,

^(Socrates) /^(Socrates) = i,

while if b is not Socrates, </>ft/t fjb — o.

To exclude this case, I should make the proviso that
must not contain “a”. To take an analogous case, put

</>x . = . x kills a, tfjx . = . x inhabits England.

Then fajita is the likelihood of a committing suicide if English,
whereas <f>xlipx, in general, is the likelihood of a being murdered
by some Englishman who is named x. Obviously, in most cases,
(jyaltjja is greater than <f>bjif)b y because a man is more likely to kill
himself than to kill another person selected at random.

The essential condition, then, seems to be that “<£#” must not
contain “a” or “ft”. If this condition is fulfilled, I do not see
how we can fail to have

(f>a/ipa = <f>b/i/jb.

I conclude that what the principle of indifference really asserts
is that probability is a relation between propositional functions,
not between propositions. This is what is meant by such phrases
as “a random selection”. This phrase means that we are to con-
sider a term solely as one satisfying a certain propositional
function; what is said is, then, really about the propositional
function and not about this or that value of it.

Nevertheless, there remains something substantial which is
what really concerns us. Given a probability- relation between two
propositional functions (f>x and tpx, we can regard this as a
relation between <f>a and ifja y provided “<£#” and do not

contain “a”. This is a necessary axiom in all applications of
probability in practice, for then it is particular cases that concern
us.

My conclusion is that the chief formal defect in Keynes's theory
of probability consists in his regarding probability as a relation
between propositions rather than between propositional functions.
The application to propositions, I should say, belongs to the uses
of the theory, not to the theory itself.

397

Chapter VI

DEGREES OF CREDIBILITY

A. GENERAL CONSIDERATIONS

T hat all human knowledge is in a greater or less degree
doubtful is a doctrine that comes to us from antiquity; it
was proclaimed by the sceptics, and by the Academy in its
sceptical period. In the modern world it has been strengthened
by the progress of science. Shakespeare, to represent the most
ridiculous extremes of scepticism, says:

Doubt that the stars are fire,

Doubt that the sun doth move.

The latter, when he wrote, had already been questioned by
Copernicus, and was about to be even more forcibly questioned
by Kepler and Galileo. The former is false, if “fire” is used in
its chemical sense. Many things which had seemed indubitable
have turned out to be in all likelihood untrue. Scientific theories
themselves change from time to time, as new T evidence accumu-
lates; no prudent man of science feels the same confidence in
a recent scientific theory as was felt in the Ptolemaic theory
throughout the middle ages.

But although every part of what we should like to consider
“knowledge” may be in some degree doubtful, it is clear that
some things are almost certain, while others are matters of
hazardous conjecture. For a rational man, there is a scale of
doubtfulness, from simple logical and arithmetical propositions,
and perceptive judgments, at one end, to such questions as what
language the Myceneans spoke or “what song the Sirens sang”,
at the other. Whether any degree of doubtfulness attaches to the
least dubitable of our beliefs, is a question with which we need
not at present concern ourselves ; it is enough that any proposition
concerning which we have rational grounds for some degree of
belief or disbelief can, in theory, be placed in a scale between
certain truth and certain falsehood. Whether these limits are
themselves to be included, we may leave an open question.

There is a certain connection between mathematical probability

398

DEGREES OF CREDIBILITY

and degrees of credibility. The connection is this: When, in
relation to all the available evidence, a proposition has a certain
mathematical probability, then this measures its degree of credi-
bility. For instance, if you are about to throw dice, the proposition
“double sixes will be thrown” has only one thirty-fifth of the
credibility attaching to the proposition “double sixes will not be
thrown”. Thus the rational man, who attaches to each proposition
the right degree of credibility, will be guided by the mathematical
theory of probability when it is applicable .

The concept “degree of credibility”, however, is applicable
much more widely than that of mathematical probability; I hold
that it applies to every proposition except such as neither are
data nor are related to data in any way which is favourable or
unfavourable to their acceptance. I hold, in particular, that it
applies to propositions that come as near as is possible to merely
expressing data. If this view is to be logically tenable, we must hold
that the degree of credibility attaching to a proposition is itself
sometimes a datum. I think we should also hold that the degree
of credibility to be attached to a datum is sometimes a datum,
and sometimes (perhaps always) falls short of certainty. We may
hold, in such a case, that there is only one datum, namely, a
proposition with a degree of credibility attached to it, or we may
hold that the datum and its degree of credibility are two
separate data. I shall not consider which of these two views
should be adopted.

A proposition which is not a datum may derive credibility from
various different sources ; a man who wishes to prove his innocence
of a crime may argue both from an alibi and from his previous
good character. The grounds in favour of a scientific hypothesis
are practically always composite. If it is admitted that a datum
may not be certain, its degree of credibility may be increased
by an argument, or, on the contrary, may be rendered very small
by a counter-argument.

The degree of credibility conferred by an argument is not
capable of being estimated simply. Take, first, the simplest
possible case, namely that in which the premisses are certain and
the argument, if valid, is demonstrative. At each step we have to
“see” that the conclusion of this step follows from its premisses.
Sometimes this is easy, for example if the argument is a syllogism
in Barbara. In such a case, the degree of credibility attaching to

399

HUMAN knowledge: its scope and limits

the connection of premisses and conclusion is almost certainty,
and the conclusion has almost the same degree of credibility as
the premisses. But in the case of a difficult mathematical argument
the chance of an error in reasoning is much greater. The logical
connection may be completely obvious to a good mathematician,
while to a pupil it is barely perceptible, and that only at moments.
The pupil’s grounds for believing in the validity of the step are
not purely logical; they are in part arguments from authority.
These arguments are by no means demonstrative, for even the
best mathematicians sometimes make mistakes. On such grounds,
as Hume points out, the conclusion of a long argument has less
certainty than the conclusion of a short one, for at each step there
is some risk of error.

By means of certain simplifying hypotheses, this source of
uncertainty could be brought within the scope of the mathematical
theory of probability. Suppose it established that, in a certain
branch of mathematics, good mathematicians are right in a step
in their arguments in a proportion x of all cases ; then the chance
that they are right throughout an argument of n steps is It
follows that a long argument which has not been verified by
repetition runs an appreciable risk of error, even if x is nearly i.
But repetition can reduce the risk until it becomes very small. All
this is within the scope of the mathematical theory.

What, however, is not within the scope of that theory is the
private conviction of the individual mathematician as he takes
each step. This conviction will vary in degree according to the
difficulty and complexity of the step ; but in spite of this variability
it must be as direct and immediate as our confidence in objects
of perception. To prove that a certain premiss implies a certain
conclusion, we must “see” each step ; we cannot prove the validity
of the step except by breaking it up into smaller steps, each of
which will then have to be “seen”. Unless this is admitted, all
arguments will be lost in an endless regress.

I have been speaking, so far, of demonstrative inference, but
as regards our present question non-demonstrative inference
presents no new problem, for, as we have seen, even demonstrative
inference, when carried out by human beings, only confers
probability on the conclusion. It cannot even be said that reasoning
which professes to be demonstrative always confers a higher
degree of probability on the conclusion than reasoning which is

DEGREES OF CREDIBILITY

avowedly only probable; of this there are many examples in
traditional metaphysics.

If — as I believe, and as I shall argue in due course — data, as
well as results of inference, may be destitute of the highest
attainable degree of credibility, the epistemological relation
between data and inferred propositions becomes somewhat com-
plex. I may, for instance, think that I recollect something, but
find reason to believe that what I seemed to recollect never
happened; in that case I may be led by argument to reject a
datum. Conversely, when a datum has, per se, no very high degree
of credibility, it may be confirmed by extraneous evidence; for
example, I may have a faint memory of dining with Mr. So-and-So
some time last year, and may find that my diary for last year has
an entry which corroborates my recollection. It follows that every
one of my beliefs may be strengthened or weakened by being
brought into relation with other beliefs.

The relation between data and inferences, however, remains
important, since the reason for believing no matter what must
be found, after sufficient analysis, in data, and in data alone. (I
am here including among data the principles used in any in-
ferences that may be involved.) What does result is that the data
relevant to some particular belief may be much more numerous
than they appear to be at first sight. Take again the case of
memory. The fact that I remember an occurrence is evidence,
though not conclusive evidence, that the occurrence took place.
If I find a contemporary record of the occurrence, that is con-
firmatory evidence. If I find many such records, the confirmatory
evidence is strengthened. If the occurrence is one which, like a
transit of Venus, is made almost certain by a well-established
scientific theory, this fact must be added to the records as an
additional ground for confidence. Thus while there are beliefs
which are only conclusions of arguments, there are none which,
in a rational articulation of knowledge, are only premisses. In
saying this, I am speaking in terms of epistemology, not of logic.

Thus an epistemological premiss may be defined as a proposition
which has some degree of rational credibility on its own account,
independently of its relations to other propositions. Every such
proposition can be used to confer some degree of credibility on
propositions which either follow from it or stand in a probability
relation to it. But at each stage there is some diminution of the

401 CC

human knowledge: its scope and limits

original stock of credibility; the case is analogous to that of a
fortune which is lessened by death duties on each occasion when
it is inherited. Carrying the analogy a little further, we may say
that intrinsic credibility is like a fortune acquired by a man’s
own efforts, while credibility as the result of an argument is like
inheritance. The analogy holds in that a man who has made a
fortune can also inherit one, though every fortune must owe its
origin to something other than inheritance.

In this chapter I propose to discuss credibility, first in relation
to mathematical probability, then in relation to data, then in
relation to subjective certainty, and finally in relation to rational
behaviour.

B. CREDIBILITY AND FREQUENCY

I am now concerned to discuss the question: In what circum-
stances is the credibility of a proposition a derived from the
frequency of ifix given some <j>x ? In other words, if “<£a” is
“a is an a”, in what circumstances is the credibility of
“ a is a /8” derived from one or more propositions of the form:
“A proportion m/n of the members of a are members of /3”? This
question, we shall find, is not quite so general as the one we
ought to ask, but it will be desirable to discuss it first.

It seems clear to common sense that, in the typical cases of
mathematical probability, it is equal to degree of credibility. If
I draw a card at random from a pack, the degree of credibility
of “the card will be red” is exactly equal to that of “the card will
not be red”, and therefore the degree of credibility of either is 1/2,
if 1 represents certainty. In the case of a die, the degree of credi-
bility of “1 will come uppermost” is exactly the same as that of
“2 will come uppermost”, or 3 or 4 or 5 or 6. Hence all the
derived frequencies of the mathematical theory can be interpreted
as derived degrees of credibility.

In this translation of mathematical probabilities into degrees
of credibility, we make use of a principle which the mathematical
theory does not need. The mathematical theory merely counts
cases; but in the translation we have to know, or assume, that
each case is equally credible. The need of this principle has long
been recognized ; it has been called the principle of non-sufficient
reason, or (by Keynes) the principle of indifference. We COn-

'tfi .

402

DEGREES OF CREDIBILITY

sidered this principle in connection with Keynes, but we must
now consider it on its own account. Before discussing it, I wish
to point out that it is not needed in the mathematical theory of
probability. In that theory, we only need to know the numbers
of various classes. It is only when mathematical probability is
taken as a measure of credibility that the principle is required.

What we need is something like the following: “Given an
object a , concerning which we wish to know what degree of
credibility to attach to the proposition ‘ a is a and given that
the only relevant knowledge we have is { a is an a, then the
degree of credibility of ‘ a is a 0’ is the mathematical probability
measured by the ratio of the number of members common to a
and P to the number of members of a”.

Let us illustrate this by considering once more the tallest
person in the United States, and the chance that he lives in Iowa.
We have here, on the one hand, a description d y known to be
applicable to one and only one of a number of named persons
A x , A 2 , . . . A„, where n is the number of inhabitants of the United
States. That is to say, one and only one of the propositions
“ d = A,” (where r runs from i to n) is known to be true, but
we do not know which. If this is really all our relevant knowledge,
we assume that any one of the propositions “ d = A r ” is as
credible as any other. In that case, each has a credibility i jn.
If there are m inhabitants of Iowa, the proposition “ d inhabits
Iowa” is equivalent to a disjunction of m of the propositions
“ d = A/’, and therefore has m times the credibility of any one
of them, since they are mutually exclusive. Therefore it has a
degree of credibility measured by mjn.

Of course in the above illustration the propositions “ d = A”
are not all on a level. The evidence enables us to exclude children
and dwarfs, and probably women. This shows that the principle
may be difficult to apply, but does not show that it is false.

The case of drawing a card from a pack comes nearer to
realizing the conditions required by the principle. Here the
description “d” is “the card I am about to draw”. The 52 cards
all have what we may regard as names: “2 of spades”, etc. We
have thus 52 propositions “</ — A r ”, of which one and only one
is true, but we have no evidence whatever inclining us to one
rather than another. Therefore the credibility of each is 1 /s 2. This,
if admitted, connects credibility with mathematical probability.

403

HUMAN KNOWLEDGE: ITS SCOPE AND LIMITS

We may therefore enunciate, as a possible form of the “principle
of indifference”, the following axiom :

“Given a description d , concerning which we know that it is
applicable to one and only one of the objects a lt a 2i . . . a nt and
given that we have no knowledge bearing on the question which
of these objects the description applies to, then the n propositions
*d= a r * ( i < r < n) are all equally credible, and therefore each
has a credibility measured by i /«.”

This axiom is more restricted than the principle of non-
sufficient reason as usually enunciated. We have to inquire
whether it will suffice, and also whether we have reason to
believe it.

Let us first compare the above with Keynes’s principle of
indifference, discussed in an earlier chapter. It will be re-
membered that his principle says: the probabilities of p and q
relative to given evidence are equal if (i) the evidence is
symmetrical with respect to p and q , (2) p and q are “indi-
visible”, i.e. neither is a disjunction of propositions of the
same form as itself. We decided that this could be simplified:
what is needed, we said, is that p and q should be values of one
propositional function — say p— (j>(a) and q = <f>(b ) ; that
should not contain either a or b\ and that, if the evidence contains
a mention of a , say in the form ip(a) y it must also contain i/j(b) y
and vice versa, where «/oc, in turn, must not mention a or b. This
principle is somewhat more general than the one enunciated in
the previous paragraph : it implies the latter, but I doubt whether
the latter implies it. We may perhaps accept the more general
principle, and re-state it as follows:

“Given two propositional functions <f>x, ifjx , neither of which
mentions a or 6, or, if it does so, mentions them symmetrically,
then, given ipa and tpb, the two propositions 4 >a, <f>b have equal
credibility”.

This principle, if accepted, enables us to infer credibility from
mathematical probability, and makes all the propositions of the
mathematical theory available for measuring degrees of credibility
in the cases to which the mathematical theory is applicable.

Let us apply the above principle to the case of n balls in a
bag, each of which is known to be either white or black; the
question is: what is the probability that there are x white balls?
Laplace assumed that every value of x from o to n is equally

404

DEGREES OF CREDIBILITY

likely, so that the probability of a given x is i /(« + i). From
a purely mathematical standpoint, this is legitimate, provided
we start from the propositional function :

x = the number of white balls.

But if we start from the propositional function :

x is a white ball,

we obtain a quite different result. In this case, there are many
ways of choosing x balls. The first ball can be chosen in n ways;
when it has been chosen, the next can be chosen in n — i ways,
and so on. Thus the number of ways of choosing x balls is

n times (w — i) times ( n — 2) times . . . times (n — x + 1).
This is the number of ways in which there can be x white balls.
To get the probability of x white balls, we have to divide this
number by the sum of the numbers of ways of choosing o white
balls, or 1, or 2, or 3, or . . . or n. This sum is easily shown to
be 2 W . Therefore the chance of exactly x white balls is obtained
by dividing the above number by 2 W . Let us call it “/>(w, r)”.

This has a maximum when x = \n if n is even, or when
x = \n ± \ if n is odd. Its value when x or n — x is small is
very small if n is large. From the purely mathematical point of
view, these two very different results are equally legitimate. But
when we come to the measurement of degrees of credibility, there
is a great difference between them. Let us have some way, inde-
pendent of colour, by which we can distinguish the balls ; e.g. let
them be successively drawn out of the bag, and let us call the
one first drawn d u the one drawn second </ 2 > and so on. Put “0”
for “white” and “A” for “black”, and put “<£0” for “white is
the colour of d^\ “<£&” for “black is the colour of d The
evidence is that cf>a or <f>b is true, but not both. This is symmetrical,
and therefore, on the evidence, <j>a and <f>b have equal credibility,
i.e. “dj is white” and “d { is black” have equal credibility. The
same reasoning applies to d 2y rf 3 , . . . d n . Thus in the case of each
ball the degrees of credibility of white and black are equal. And
therefore, as a simple calculation shows, the degree of credibility
of * white balls is p(n , x ), where it is assumed that x lies between
o and w, both included.

It is to be observed that, in measuring degrees of credibility,
we suppose the data not only true, but exhaustive in relation to

405

human knowledge: its scope and limits

our knowledge, i.e. we assume that we know nothing relevant
except what is mentioned in the data. Therefore for a given person
at a given time there is only one right value for the degree of
credibility of a given proposition, whereas in the mathematical
theory many values are equally legitimate in relation to many
different data, which may be purely hypothetical.

In applying the results of the mathematical calculus of proba-
bility to degrees of credibility, we must be careful to fulfil two
conditions. First, the cases which form the basis of the mathe-
matical enumeration must all be equally credible on the evidence;
second, the evidence must include all our relevant knowledge.
As to the former of these conditions a few words must be said.

Every mathematical calculation of probability starts from some
fundamental class, such as a certain number of tosses of a coin,
a certain number of throws of a die, a pack of cards, a collection
of balls in a bag. Each member of this fundamental class counts
as one. From it we manufacture other logically derivative classes,
e.g. a class of n series of ioo tosses of a coin. Out of these n series
we can pick out the sub-class of those that consist of 50 heads
and 50 tails. Or, starting from a pack of cards, we can consider
the class of possible “hands”, i.e. selections of 13 cards, and
proceed to inquire how many of these contain 11 cards of one
suit. The point is that the frequencies that are calculated always
apply to classes having some structure logically defined in relation
to the fundamental class, whereas the fundamental class, for the
purposes of the problem, is regarded as composed of members
having no logical structure, i.e. their logical structure is irrelevant.

So long as we confine ourselves to the calculation of frequencies,
i.e. to the mathematical theory of probability, we can take any
class as our fundamental class, and calculate frequencies in relation
to it. It is not necessary to make an assumption to the effect that all
the members of the class are equally probable; all that we need
to say is that, for the purpose in hand, each member of the class
is to count as one. But when we wish to ascertain degrees of
credibility, it is necessary that our basic class should consist of
propositions which are all equally credible in relation to the
evidence. Keynes’s “indivisibility” is intended to secure this. I
should prefer to say that the members of the fundamental class
must have “relative simplicity”, i.e. they must not have a structure
definable in terms of the data. Take, e.g., white and black balls

406

DEGREES OF CREDIBILITY

in a bag. Each ball has, in fact, an incredibly complicated struc-
ture, since it consists of billions of molecules; but this is quite
irrelevant to our problem. On the other hand, a collection of m
balls chosen from a fundamental class of n balls has a logical
structure relatively to the fundamental class. If each member of
the fundamental class has a name, every sub-class of m terms
can be defined. All calculations of probability have to do with
classes which can be defined in terms of the fundamental class.
But the fundamental class itself must consist of members which
cannot be logically defined in terms of the data. I think that when
this condition is fulfilled the principle of indifference is always
satisfied.

At this point, however, a caution is necessary. There are two
ways in which “a is an a” may become probable, either (i) because
it is certain that a belongs to a class most of which are a’s, or
(2) because it is probable that a belongs to a class all of which
are a’s. For instance, we may say “Mr. A. is probably mortal”
if we are sure that most men are mortal, or if we have reason
to think it probable that all men are mortal. When we make a
throw with two dice, we can say “probably we shall not throw
double sixes”, because we know that most throws are not double
sixes. On the other hand, suppose I have evidence suggesting,
but not proving, that a certain bacillus is always present in a
certain disease; I may then say, in a given case of this disease,
that probably the bacillus in question is present. There is in each
case a kind of syllogism. In the first case,

Most A is B ;

This is an A;

Therefore this is probably a B.

In the second case,

Probably all A is B ;

This is an A ;

Therefore this is probably a B.

The second case, however, is more difficult to reduce to a fre-
quency. Lgt us inquire whether this is possible.

In some cases, this is clearly possible. E.g. most \Vords do not
contain the letter Z. Therefore, if some word is chosen at hap-
hazard, it is probable that all its letters are other than Z. Thus if

407

HUMAN knowledge: its scope and limits

A = the class of letters in the word in question, and B = the
class of letters other than Z, we get a case of our second pseudo-
syllogism. The word, of course, must be defined in some way
which leaves us in temporary ignorance as to what it is, e.g. the
8,ooo** word in Hamlet , or the third word on p. 248 of the
Concise Oxford Dictionary . Assuming that you do not at present
know what these words are, you will be wise to bet against their
containing a Z.

In all cases of our second pseudo-syllogism, it is clear that
what I have been calling the “fundamental class” is given as
a class of classes, and therefore its logical structure is essential.
To generalize the above instance : let k be a class of classes, such
that most of its members are entirely contained in a certain
class / 3 ; then, from *‘x is an a” and “a is a we can conclude
“ x is probably a £”. (In the above instance, k was the class of
words, a the class of letters in a certain word, and the alphabet
without Z.) The odd thing is that, denoting by “sum of #c” the
class of members of members of k, our premisses do not suffice
to prove that a member of the sum of k is probably a member
of j8. For example, let k consist of the three words STRENGTH,
QUAIL, MUCK, together with all words containing no letter
occurring in any of these three. Then the sum of k consists of
all the letters of the alphabet, possibly excepting Z. 1 But “ x is
an a and a is a /c” makes it probable that x is not one of the
letters occurring in the above three words, while “ x is a member
of the sum of k” does not make this probable. This illustrates
the complications that arise when the fundamental class has a
structure which is relevant to the probabilities. But in such cases
as the above it is still possible to measure credibility by frequency,
though less simply.

There is, however, another and more important class of cases,
which we cannot adequately discuss except in connection with
induction. These are the cases where we have inductive evidence
making it probable that all A is B, and we infer that a particular
A is probably a B, e.g. probably all men are mortal ( not all men
are probably mortal), therefore Socrates is probably mortal. This
is a pseudo-syllogism of our second kind. But if the “probably”
in “probably all men are mortal” can be reduced to a frequency,

1 Whether Z is to be included depends upon whether “ZOO” is
allowed to count as a word.

408

DEGREES OF CREDIBILITY

it certainly cannot be so reduced at all simply. I will therefore
leave this class of cases to be discussed at a later stage.

There are, we shall find, various examples of degrees of credi-
bility not derivable from frequencies. These I shall now proceed
to consider.

C. CREDIBILITY OF DATA

In the present section I propose to advocate an unorthodox
opinion, namely, that a datum may be uncertain. There have been
hitherto two views: first, that in a proper articulation of know-
ledge we start from premisses which are certain in their own
right, and may be defined as “data”; second, that, since no know-
ledge is certain, there are no data, but our rational beliefs form
a closed system in which each part lends support to every other
part. The former is the traditional view, inherited from the

Greeks, enshrined in Euclid and theology; the latter is a view

first advocated, if I am not mistaken, by Hegel, but most

influentially supported, in our day, by John Dewey. The view

which I am about to set forth is a compromise, but one somewhat
more in favour of the traditional theory than of that advocated
by Hegel and Dewey.

I define a “datum” as a proposition which has some degree
of rational credibility on its own account, independently of any
argument derived from other propositions. It is obvious that the
conclusion of an argument cannot derive from the argument a
higher degree of credibility than that belonging to the premisses ;
consequently, if there is such a thing as rational belief, there
must be rational beliefs not wholly based on argument. It does
not follow that there are beliefs which owe none of their credibility
to argument, for a proposition may be both inherently credible
and also a conclusion from other propositions that are inherently
credible. But it does follow that every proposition which is
rationally credible in any degree must be so either (a) solely in
its own right, or (6) solely as the conclusion from premisses which
are rationally credible in their own right, or (c) because it has
some degree of credibility in its own right, and also follows, by
a demonstrative or probable inference, from premisses which
have some degree of credibility in their own right. If all proposi-
tions which have any credibility in their own right are certain,
case (c) has no importance, since no argument can make such

409

human knowledge: its scope and limits

propositions more certain. But on the view which I advocate,
case (c) is of the greatest importance.

The traditional view is adopted by Keynes, and set forth by
him in his Treatise on Probability , p. 16. He says:

“In order that we may have a rational belief in p of a lower
degree of probability than certainty, it is necessary that we know
a set of propositions A, and also know some secondary proposition
q asserting a probability- relation between p and A.

“In the above account one possibility has been ruled out. It
is assumed that we cannot have a rational belief in p of a degree
less than certainty except through knowing a secondary proposi-
tion of the prescribed type. Such belief can only arise, that is
to say, by means of the perception of some probability- relation.
. . . All knowledge which is obtained in a manner strictly direct
by contemplation of the objects of acquaintance and without any
admixture whatever of argument and the contemplation of the
logical bearing of any other knowledge on this, corresponds to
certain rational belief and not to a merely probable degree of
rational belief.”

I propose to controvert this view. For this purpose I shall
consider (1) faint perception, (2) uncertain memory, (3) dim
awareness of logical connection.

(1) Faint perception . — Consider such familiar experiences as
the following, (a) You hear an aeroplane going away; at first
you are sure you hear it, and at last you are sure you do not
hear it, but in the interval there is a period during which you are
not sure whether you still hear it or not. ( b ) You are watching
Venus during the dawn ; at first you see the planet shining brightly,
and at last you know that daylight has made it invisible, but
between these two times you may be in doubt whether you are
still seeing it or not. (c) In the course of travel you have attracted
a number of fleas; you set to work to get rid of them, and in
the end you are sure you have succeeded, but in the meantime
you are troubled by occasional doubtful itches. ( d ) By mistake
you make tea in a pot that has contained vinegar; the result is
appalling. You rinse the pot and try again, but still the offensive
flavour is unmistakeable. After a second rinsing you are doubtful
whether you still taste the vinegar; after a third you are sure
you do not. (e) Your drains are out of order, and you call in
the plumber. At first, after his visit, you feel sure that the offensive

410

DEGREES OF CREDIBILITY

odour is gone, but gradually, through varying stages of doubt, you
become certain that it has returned.

Such experiences are familiar to every one, and must be taken
account of in any theory as to the knowledge based on sense-
perception.

(2) Uncertain memory .— In The Tempest (Act I, Scene II),
Prospero asks Miranda to look into “the dark backward and
abysm of time”; she says “had I not four or five women once
that tended me?” and Prospero confirms her doubtful recollec-
tion. We all have memories of this kind, about which we do not
feel sure. Usually, if it is worth while, we can discover from other
evidence whether they are veridical or not, but that is irrelevant
to our present thesis, which is that they have a certain degree
of credibility on their own account, though this degree may fall
far short of full certainty. A recollection which has a fairly high
degree of credibility contributes its quota to our grounds for
believing in some past occurrence for which we have other
evidence. But here a distinction is necessary. The past event I
uncertainly remembered has partial credibility in itself ; but when
I adduce the recollection as a ground for belief, I am no longer
treating the past occurrence as a datum, for it is not it but the
present recollecting that is my datum. My recollecting confers
some credibility on what is recollected; how much credibility,
we can more or less ascertain inductively by a statistical inquiry
into the frequency of errors of memory. But this is a different
matter from past occurrences as data. That such data must be
supplied by memory is a thesis which I have argued elsewhere.

(3) Dim awareness of logical connection . — Any person whose
mathematical abilities are not almost superhuman must, if he has
studied mathematics, have often had the experience of being
hardly able to “see” a certain step in a proof. The process of
following a proof is facilitated by making the steps very small,
but however small we make them some of them may remain
difficult if the subject-matter is very complex. It is obvious that,
if we have made the steps as small as possible, each step must
be a datum, for otherwise every attempt at proof would involve
an endless regress. Consider, say, a syllogism in Barbara. I say
“all men are mortal”, and you agree. I say “Socrates is a man”,
and you agree. I then say “therefore Socrates is mortal”, and
you say “I don't see how that follows”. What, then, can I do?

41 1

HUMAN knowledge: its scope and limits

I can say: “Don’t you see that if f(x) is always true, then f(a)
is true? and don’t you see that therefore if <f>(x) always implies
t/f(x ), then <f> (Socrates) implies 0 (Socrates)? and don’t you see
that I can put 4 x is a man’ for 4 0#’ and l x is mortal’ for 4 0#’?
And don’t you see that this proves my point?” A pupil who could
follow this but not the original syllogism would be a psychological
monstrosity. And even if there were such a pupil, he would still
have to “see” the steps of my new argument.

It follows that, when an argument is stated as simply as possible,
the connection asserted in every step has to be a datum. But it
is impossible that the connection in every step should have the
highest degree of credibility, because even the best mathematicians
sometimes mate mistakes. In fact, our perceptions of the logical
connections between propositions, like our sense-perceptions and
our memories, can be ordered by their degrees of credibility:
in some, we see the logical connection so clearly that we cannot
be made to doubt it, while in others our perception of the con-
nection is so faint that we are not sure whether we see it or not.

I shall henceforth assume that a datum, in the sense defined
at the beginning of this section, may be uncertain in a greater
or less degree. We can, theoretically, make a connection between
this kind of uncertainty and the kind derived from mathematical
probability, if we suppose that an uncertainty of one kind can
be judged greater than, equal to, or less than, one of the other
kind. For example, when I think I hear a faint sound, but am
not sure, I may theoretically be able to say: The occurrence of
this sound has the same degree of rational credibility as the
occurrence of double sixes with dice. In some degree, such com-
parisons could be tested, by collecting evidence of mistakes as
to faint sensations and working out their frequency. All this is
vague, and I do not see how to make it precise. But at any rate
it suggests that the uncertainty of data is quantitative, and can
be equal or unequal to the uncertainty derived from a probability
inference. I shall assume this to be the case, while admitting that,
in practice, the numerical measurement of the uncertainty of a
datum is seldom possible. We may say that the uncertainty is a
half when the doubt is such as to leave an even balance between
belief and disbelief. But such a balance can only be established
by introspection, and is incapable of being confirmed by any sort
of test.

412

DEGREES OF CREDIBILITY

The admission of uncertainty in data complicates the process
of estimating the rational credibility of a proposition. Let us
suppose that a certain proposition p has a degree of credibility x
on its own account, as a datum; and let us suppose that there
is also a conjunction h of propositions, having intrinsic credibility jy,
from which it follows, by an argument having credibility s, that p
has a degree of credibility w. What, then, is the total credibility
of p ? Perhaps we might be inclined to say that it is x + yzw.
But h also is sure to have a derived as well as an intrinsic credibility,
and this will increase the credibility of x . In fact, the complications
will soon become unmanageable. This causes a certain approxi-
mation to the theory of Hegel and Dewey.

Given a number of propositions, each having a fairly high
degree of intrinsic credibility, and given a system of inferences
by virtue of which these various propositions increase each other ’s
credibility, it may be possible in the end to arrive at a body of
interconnected propositions having, as a whole, a very high degree
of credibility. Within this body, some are only inferred, but none
are only premisses, for those which are premisses are also con-
clusions. The edifice of knowledge may be compared to a bridge
resting on many piers, each of which not only supports the road-
way but helps the other piers to stand firm owing to intercon-
necting girders. The piers are the analogues of the propositions
having some intrinsic credibility, while the upper portions of the
bridge are the analogues of what is only inferred. But although
each pier may be strengthened by the other piers, it is the solid
ground that supports the whole, and in like manner it is intrinsic
credibility that supports the whole edifice of knowledge.

D. DEGREES OF SUBJECTIVE CERTAINTY

Subjective certainty is a psychological concept, while credibility
is at least in part logical. The question whether there is any
connection between them is a form of the question whether we
know anything. Such a question cannot be discussed on a basis
of complete scepticism; unless we are prepared to assert something ,
no argument is possible.

Let us first distinguish three kinds of certainty .

(i) A propositional function is certain with respect to another
when the class of terms satisfying the second is part of the class

413

HUMAN knowledge: its scope and limits

of terms satisfying the first. E.g. “x is an animal” is certain in
relation to “ x is a rational animal”. This meaning of certainty
belongs to mathematical probability. We will call this kind of
certainty “logical”.

(2) A proposition is certain when it has the highest degree of
credibility, either intrinsically or as a result of argument. Perhaps
no proposition is certain in this sense, i.e. however certain it
may be in relation to a given person’s knowledge, further know-
ledge might increase its de

Internet Archive Audio

Featured

Top

Images

Featured

Top

Software

Featured

Top

Books

Featured

Top

Video

Featured

Top

Mobile Apps

Browser Extensions

Archive-It Subscription

Save Page Now

Full text of "Human Knowledge - Its Scope And Limits"

See other formats